CD 

a 

Cvl 

o 


Irving  Stringham 

—-.^.^  ^ • ijj^^asrsssraaasasasai 


A  TREATISE 


ANALYTIC  GEOMETRY, 


ESPECIALLY   AS   APPLIED   TO 


THE  PROPERTIES  OF  CONICS: 


INCLUDING  THE  MODERN  METHODS  OF  ABRIDGED  NOTATION. 


WRITTEN   FOR   THE   MATHEMATICAL   COURSE  OF 

JOSEPH  KAY,  M.D., 


BY 

GEORGE  H.  HOWISON,  M.A., 

PROFESSOR    IN    WASHINGTON    UNIVERSITY. 


CINCINNATI: 
WILSON,    HIISTKLE    <fc    CO. 

PHIL'A :  CLAXTON,  REMSEN  &  HAFFELPINGER. 
NEW  YORK:  CLARK  &  MAYNARD. 


Entered,  according  to  Act  of  Congress,  in  the  year  1869;  by 
WILSON,    HINKLE    &    CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the 
Southern  District  of  Ohio. 


ELECTBOTYPED  AT  THE 

FBANKLIN    TYPE     FOUNDRY, 

CINCINNATI. 


PREFACE. 


IN  preparing  the  present  treatise  on  Analytic  Geometry,  I  have 
had  in  view  two  principal  objects :  to  furnish  an  adequate  intro- 
duction to  the  writings  of  the  great  masters;  and  to  produce  a 
book  from  which  the  topics  of  first  importance  may  readily  be 
selected  by  those  who  can  not  spare  the  time  required  for  reading 
the  whole  work.  I  have  therefore  presented  a  somewhat  ex- 
tended account  of  the  science  in  its  latest  form,  as  applied  to 
Loci  of  the  First  and  Second  Orders ;  and  have  endeavored  to 
perfect  in  the  subject-matter  that  natural  and  scientific  arrange- 
ment which  alone  can  facilitate  a  judicious  selection. 

Accordingly,  not  only  have  the  equations  to  the  Eight  Line, 
the  Conies,  the  Plane,  and  the  Quadrics  been  given  in  a  greater 
variety  of  forms  than  usual,  but  the  properties  of  Conies  have 
been  discussed  with  fullness;  and  the  Abridged  Notation  has 
been  introduced,  with  its  cognate  systems  of  Trilinear  and  Tan- 
gential Co-ordinates.  On  the  other  hand,  to  facilitate  selection, 
these  modern  methods  have  been  treated  in  separate  chapters; 
and,  in  the  discussion  of  properties,  distinct  statement,  as  well  as 
natural  grouping,  has  been  constantly  kept  in  view. 

It  is  to  be  hoped,  however,  that  omissions  will  be  avoided 
rather  than  sought,  and  that  the  modern  methods,  which  are 
here  for  the  first  time  presented  to 'the  American  student,  may 
awaken  a  fresh  interest  in  the  subject,  and  lead  to  a  wider  study 
of  it,  in  the  remarkable  properties  and  elegant  forms  with  which 

(iii) 


800547 


PREFACE. 

it  has  been  enriched  in  the  last  fifty  years.  The  labors  of  PON- 
CELET,  STEINER,  MOBIUS,  and  PLUCKER  have  well-nigh  wrought 
a  revolution  in  the  science ;  and  though  the  new  properties  which 
they  and  their  followers  have  brought  to  light,  have  not  yet  re- 
ceived any  sufficient  application,  nevertheless,  in  connection  with 
the  elegant  and  powerful  methods  of  notation  belonging  to  them, 
they  constitute  the  chief  beauties  of  the  subject,  and  have  very 
much  heightened  its  value  as  an  instrument  of  liberal  culture. 

To  render  the  book  useful  as  a  work  of  reference,  has  also  been 
an  object.  In  the  Table  of  Contents,  a  very  full  synopsis  of 
properties  and  constructions  will  be  found,  which  it  is  hoped 
will  meet  the  wants,  not  only  of  the  student  in  reviewing,  but 
of  the  practical  workman  as  well. 

In  the  demonstrations,  convenience  and  elegance  have  been 
aimed  at,  rather  than  novelty.  When  it  has  seemed  preferable 
to  do  so,  I  have  followed  the  lines  of  proof  already  indicated  by 
the  leading  writers,  instead  of  striking  out  upon  fresh  ones.  My 
chief  indebtedness  in  this  respect,  is  to  the  admirable  works  of 
Dr.  GEORGE  SALMON.  The  treatise  of  Mr.  Todhunter  has  fur- 
nished some  important  hints;  while  those  of  O'Brien  and  Hymers 
have  been  often  referred  to.  For  examples,  I  have  drawn  upon 
the  collections  of  Walton,  Todhunter,  and  Salmon.  Of  American 
works,  those  of  Peirce  and  Church  have  been  consulted  with 
advantage. 

To  Professor  William  Chauvenet,  Chancellor  of  Washington 
University,  formerly  Head  of  the  Department  of  Mathematics 
in  the  United  States  Naval  Academy,  I  am  indebted  for  many 
valuable  suggestions. 

H. 

WASHINGTON  UNIVERSITY,  | 
ST.  Louis,  Sept.,  18C9.     J 


NOTE  TO  SECOND  EDITION. 


At  the  suggestion  of  several  instructors,-  I  place  here  an  OUTLINE 
OF  THE  COURSES  OF  STUDY  which  seem  to  me  most  judicious  in 
using  the  present  treatise. 

MINIMUM  COURSE. 

BOOK  I,  PART!.  ARTS.  1—6;  13—28;  46—64;  74— 
85;  95,  96,  98,  99;  101—103;  106;  133— 
138;  145—152;  165—172;  179—184. 

BOOK  II,  PART  I.  ARTS.  293—305;  310—317;  351—357; 
359—372;  376;  379—385;  389—392;  402—406; 
411—413;  416— 418;  421,  422;  427—429;  442— 
444;  446— 454;  456— 469;  473;  476—481;  485— 
488;  497—501;  506,  507,  510,  511,  514,  515; 
520—522;  535—543;  546—550;  553—557;  559— 
576;  579—586;  594-609;  622—634. 

BOOK  II.  ARTS.  674 — 690,  including  the  general  doctrine 
of  Space-coordinates  and  of  the  Plane. 

To  these  articles  there  should  be  added  such  a  selection  from  the 
Examples  as  the  Course  implies.  The  Course  will  thus  include 
about  210  pages. 

INTERMEDIATE  COURSE. 

This  Course  is  what  I  suppose  the  leading  Colleges  will  be  most 
likely  to  pursue,  and  should  therefore  include 

THE  INTRODUCTION.  ARTS.  1  —  6;   13 — 45. 

BOOK  FIRST,  PART  I.  Chapter  I. 

PART  II,  Chapter  I  to  Art.  274.  Chapter  II  to 
Art.  332,  omitting,  however,  Arts.  307,  324 — 
327.  Chapters  III — V,  omitting  Articles  in  fine 
type.  Chapter  VI  to  Art.  670. 

Boo£  SECOND.  Chapter  I.  Chapter  II,  Arts.  713,  714; 
731—741. 

THE  FULL  COURSE 

Is  intended  for  such  students  as  desire  to  make  Mathematics  a 
specialty ;  and  students  in  Schools  of  Technology  will  naturally 
read  the  whole  of  Book  Second,  even  when  they  omit  large  por- 
tions of  Book  First. 

.      THE  AUTHOR. 


ERRATA. 

Page  63,  line  20  :  for  sin  —  a>  read  sin  w. 

"    103,    "     21:  for  -f  read  =  .' 

"    103,    "     22 :  dele  the  period  at  the  close. 

"    221,    «     24:  fork"=A:8inCTe&dk"=8inA:sinC. 

"    247,    "       9  :  put  the  2  outside  of  the  brace. 

"    269,    "     28 :  for  of  read  to,  and  dele  the  period. 


CONTENTS. 


INTRODUCTION:  — THE   NATURE,   DIVISIONS,   AND 
METHOD   OF  THE   SCIENCE. 

PAGE. 

I.  DETERMINATE  GEOMETRY  : 

Principles  of  Notation, 5 

Examples,     .         .         .         . H 8 

Principles  of  Construction,        .......        8 

Examples,     .         .         .         .        .         .         .         .         .         .16 

DETERMINATE  PROBLEMS  : 

In  a  given  triangle,  to  inscribe  a  square,         .         .         .         .16 

"         "  "  "          a  rectangle  with  sides  in  given 

ratio, 17 

To  construct  a  common  tangent  to  two  given  circles,      .         .      18 

a  rectangle,  given  area  and  difference  of  sides,    .      21 

Examples,     .         .         .         .  .        .         .         .         .23 

II.  INDETERMINATE  GEOMETRY  : 

1°.   Development  of  its  Fundamental  Principle: 

The  Convention  of  Co-ordinates,         ....      26 

Distinction  between  Variables  and  Constants;    defi- 
nition of  a  Function,         .....      28 

Equations     between    co-ordinates :     their    geometric 

meaning, 29 

The  Locus  denned  and  illustrated,      .         .         .         .33 
11°.   Its  Method  outlined;   in  what  sense  it  is  Analytic: 

Manner  of  employing  geometric  equations  to  estab- 
lish properties,           .   "      .         .         .         .         .35 
Special  analytic  character  of  the  Algebraic  Calculus,      38 
Elements  of  analysis  added  by  the  Convention  of  Co- 
ordinates,  39 

III0.   Its  Divisions  and  Subdivisions: 

Algebraic  and  Transcendental  Geometry,  ...       41 
Orders  of    algebraic   loci :     Elementary   and   Higher 

Geometry,      ,.>...         .         .         .      42 

Loci  in  a  Plane  and  in  Space :  Geometry  of  Two  and 

of  Three  Dimensions, 42 

(v) 


vi  CONTENTS. 


BOOK  FIEST:  — PLANE  CO-OEDINATES. 

PART  I.   ON  THE  REPRESENTATION  OF  FORM  BY  ANALYTIC 
SYMBOLS. 

CHAPTER    FIEST. 

THE  OLDER  GEOMETRY :    BILINEAR  AND  POLAR  CO-ORDINATES. 

SECTION  I. — THE  POINT.  PAGE 

BILINEAR  OR  CARTESIAN  SYSTEM  OP  CO-ORDINATES  :    Explanation  in 

detail, 48 

Expressions  for  Point  on  either  Axis;  —  for  the  Origin,.         .      50 
POLAR  SYSTEM  OP  CO-ORDINATES :.......      52 

Expression  for  the  Pole;  —  for  Point  on  Initial  Line,  .  .  53 
Distance,  in  both  systems,  between  any  Two  Points  in  a  plane,  .  55 
Co-ordinates  of  Point  cutting  this  distance  in  a  given  ratio,  .  .  56 
TRANSFORMATION  OF  CO-ORDINATES  : 

I.  To  change  the  Origin,  Axes  remaining  parallel  to  their 

first  position, 59 

II.  To  change  the  Inclination  of  the  Axes,  Origin  remaining 

the  same, 59 

Particular   Cases:  —  1.    From   Rectangular   Axes    to 

Oblique,  ...       60 

2.  From  Oblique  to  Rectangular,      61 

3.  From   Rectangular   to   Rect- 

angular,  ....       61 

III.  To  change  System  —  from  Bilinears  to  Polars,  and  con- 
versely,   62 

IV.  To  change  the  Origin,  and  make  either  previous  Trans- 
formation at  the  same  time, 63 

GENERAL  PRINCIPLES  OF  INTERPRETATION  : 

I.  Any  single  equation  between  co-ordinates    represents  a 

Locus,    .........      65 

II.  Any  two  simultaneous  equations  represent  Determinate 

Points, 67 

III.  Any  equation  lacking  absolute  term,  represents   Locus 

passing  through  Origin,         .....      69 
IV.  Transformation  of  Co-ordinates  does  not  affect  Locus,  nor 

change  the  Degree  of  its  Equation,        ...      69 
SPECIAL    INTERPRETATION    OF    EQUATIONS  :     Tracing    their  Loci   by 

means  of  Points, 71 

Definitions  and  illustrations,     .         .         .         .         .         .         .72 

Examples:  —  Equations  to  some  of  the  Higher  Plane  Curves,      75 


CONTENTS.  vii 
SECTION  II.  —  THE  RIGHT  LINE. 

A.      THE   BIGHT    LINE    UNDER    GENERAL,    CONDITIONS. 

I.  Geometric  Point  of  View :  —  Equation  to  Right  Line  is  always  of 

the  First  Degree.  PACK. 

Equation  in  terms  of  angle  made  by  Line  with  axis  X,  and  of  its 

intercept  on  axis  Y, 79 

"                "              its  intercepts  on  the  two  axes,          ...  81 
"                "              its  perpendicular  from  the  Origin,  and  angle 

of  perp'r  with  axis  X, 83 

Polar  equation,  deduced  geometrically, 84 

II.  Analytic  Point  of  View :  —  Every  equation  of  First  Degree  in  two 

variables  represents  a  Right  Line. 

Proof  of  the  theorem  by  Algebraic  Transformation  of  the    general 

equation  of  First  Degree, 87 

Proof  by  means  of  the  Trigonometric  Function  implied  in  the  equation,  87 

Proof  by  Transformation  of  Co-ordinates,            .....  89 

Analytic  deduction  of  the  Three  Forms  of  the  equation,          .         .  92 

Reduction  of  Ax  +  By  +  C=  0  to  the  form  x  cos  a  +  y  cos  /?  —  />  =  0,    .  95 

Polar  Equation  obtained  by  Transformation  of  Co-ordinates,    .         .  97 

B.      THE   RIGHT   LINE   UNDER   SPECIAL   CONDITIONS. 

Equation-  to  Eight  Line  passing  through  Two  Fixed  Points,     .  98 
Angle    between    two    Right   Lines :    condition    that    they    shall   be 

parallel  or  perpendicular, 100 

Equation  to- Right  Line  parallel  to  given  Line;  —  perpendicular  to 

given  Line, 101 

Equation  to  Right  Line  passing  through  given  Point,  and  parallel 

to  given  Line,  ..........  103 

Equation  to  any  Right  Line  through  a  Fixed  Point,          .         .  103 
Equation  to  Right  Line  through  a  given  Point,  and  cutting  a  given 

line  at  given  angle, 105 

Equation  to  Perpendicular  through  a  given  Point,      .         .         .  106 
Length  of  Perpendicular  from  (x,y)  on  x  cos  a  4-ycos  0— p  =  0  ;   also 

on   Ax  +  By  +  C=  0, 107 

Equation  to  any  Right  Line  through  the  intersection  of  two  given  ones,  109 

Meaning  of  equation  L  +  kL'  =  0, 110 

Equation  to  Bisector  of  angle  between  any  two  Right  Lines,  .         .  113 

Equation  to  the  Right  Line  situated  at  Infinity,        ....  116 
Equations  of  Condition : 

Condition  that  Three  Points  shall  lie  on  one  Right  Line,   .         .  117 

"            "     Three  Right  Lines  shall  meet  in  One  Point,        .  118 
"            "     Movable  Right  Line  shall  pass  through  a  Fixed 

Point, 118 


Vlll  CONTENTS. 

C.      EXAMPLES   ON   THE   RIGHT   LINE. 

Examples  in  Notation  and  Conditions, 120 

Examples  of  Kectilinear  Loci, 125 

SECTION  III.  —  PAIRS  OF  EIGHT  LINES. 

I.  Geometric  Point  of  View:  —  Equation   to  a  Pair  of  Right  Lines 

is  always  of  Second  Degree. 

Formation  of  equations  in  the  type  of  LMN .  .  .  .  =  0  :  their  con- 
sequent meaning,  ........     130 

Interpretation  of  equation  LL'=  0,      ......     130 

Equation  to  Pair  of  Eight  Lines  passing  through  a  Fixed  Point,     .     131 

Meaning  of  the  equation  Axz  +  2Hxy  +  By2  =0,  .         .         .132 

Angle  between  the  Pair  Ax*  +  2Hxy  +  Bif  =  0,  .         .         .         -133 

Condition  that  they  shall  cut  at  right  angles,     ....     134 

Equation  to  Bisectors  of  angles  between  Ax"  +  2Hxy  +  By2^  0,          .     134 
Case  of  Two  Imaginary  Lines  having   Real  Bisectors   of  their 

angles, 134 

II.  Analytic   Point  of  View: — The  Equation   of  Second   Degree   in 

two  variables,   upon   a    Determinate    Condition,    represents    Two 
Right   Lines. 

Proof  of  the  theorem  by  the  mode  of  forming  LL'  =  0,     .         .         .     135 
Condition  on  which  Ax1  +  2Hxy  +  By1-  +  2Gx  +  2Fy  +  C'=  0  repre- 
sents Two  Right  Lines, 136 

SECTION  IV.  —  THE  CIRCLE. 

I.  Geometric  Point  of  View :  —Equation  to  Circle  is  always  of  Sec- 

ond Degree. 

Equation  to  the  Circle,  referred  to  any  Rectangular  Axes,  deduced 

from  geometric  definition,  .         .     138 

"  "  "  "  Oblique  Axes,     .         .         .         .139 

"  "  "  "  Rectangular  Axes  with  Origin  at 

Center, 139 

"  "  "  "  Diameter    and    Tangent    at    its 

extremity,            ....     140 
Polar  Equation  to  the  Circle, 140 

II.  Analytic  Point  of  View :  —  The  Equation  of  Second  Degree  in  two 

variables,  upon  a  Determinate  Condition,  represents  a  Circle. 

Proof  of  theorem  by  comparison  of  the  General  Equation  with  that 

to  Circle, 142 


CONTENTS.  ix 

PAGE, 

Condition  that  Ax2  +  IHxj  +  /?/  +  2Gx  +  2Fy  +  C'=  Q  shall  rep- 
resent a  Circle, 143 

To   determine   Magnitude   and   Position   of  Circle,  given   its    equa- 
tion,    144 

Condition  that  a  Circle  shall  touch  the  Axes,    .         .         .         .     /    .  145 

Examples, 146 

SECTION  V. — THE  ELLIPSE. 

I.  Geometric  Point  of  View:  —Equation   to  Ellipse  is  always  of  Sec- 

ond Degree. 

Equation  to  the  Ellipse  deduced  from  geometric  definitions,      .         .  149 

Its  general  Form,  referred  to  Axes  of  Curve  and  Focal  Center,  .  150 
Center  of  a  Curve  defined :   proof  that  Focal  Center  is  center  of  the 

Ellipse, 151 

Polar  Equation  to  Ellipse,  Center  being  Pole, 151 

"  "  "        Focus       "        " 153 

II.  Analytic  Point  of  View: — The  Equation  of  Second  Degree  in  two 

variables,  upon  a  Determinate  Condition,  represents  an  Ellipse. 

Reduction  of  Ax2  +  2Hxy  +  Bif  +  2Gx  +  2Fy+  C=  0  to  Center  of  its 

Locus,        ...........    155 

Condition  that  it  represent  an  Ellipse  is  Hz—  AB<0,      .         .         .159 
The  Point,  as  intersection  of  Two  Imaginary  Right  Lines,  a  partic- 
ular case  of  the  Ellipse, 161 

Examples, 165 

SECTION  VI. — THE  HYPERBOLA. 

I.  Geometric  Point  of  View:  —  Equation  to  Hyperbola   is  always  of 

Second  Degree. 

Equation  to  the  Hyperbola  deduced  from  geometric  definitions,  .  169 

Its  general  Form,  referred  to  the  Axes  and  Focal  Center,  .  170 

Proof  that  the  Focal  Center  is  the  center  of  the  Hyperbola,  .  172 

Polar  Equation  to  Hyperbola.  Center  being  Pole,  ....  172 

"  "  "  "  Focus  "  "  ....  174 

II.  Analytic  Point  of  View  : — The  Equation  of  Second  Degree  in  two 
variables,  upon  a  Determinate  Condition,  represents  an  Hyperbola. 

Equation  to  Hyperbola  compared  with  Reduced  Equation  of  Second 

Degree, .  175 

Condition  that  the  latter  shall  represent  an  Hyperbola  is 

H*-AB>Q, 176 

Two  Right  Lines  intersecting,  a  particular  case  of  the  Hyperbola,    .     177 


x  CONTENTS. 

PAGE. 

Examples  on  the  Hyperbola, 178 

SECTION  VII.— THE  PARABOLA. 

I.  Geometric  Point  of  View:  —  Equation  to  Parabola   is  always   of 

Second  Degree. 

Equation  to  Parabola  deduced  from  geometric  definitions,         .         .     181 

Its  general  Form,  referred  to  Axis  and  Directrix,       .         .         .     182 

Polar  Equation  to  Parabola,  Focus  being  Pole,  ....     183 

II.  Analytic  Point  of  View: — The  Equation  of  Second  Degree  in  two 

variables,  upon  a  Determinate  Condition,  represents  a  Parabola. 

Additional  transformation  of  General  Equation,  under  condition  of 

non-centrality, 185 

Condition  that  it  represent  a  Parabola  is  H2  —  AS=  0,     .         .         .  188 
The  Right  Line  as  Center  of  Two  Parallels,        .         .         .         .189 
"         "         "      as  Limit  of  Two  Parallels,  a  particular  case  of 

the  Parabola, 190 

Examples, 191 

SECTION  VIII.  —  Locus  OF  SECOND  ORDER  IN  GENERAL. 

Summary   of  Conditions  already  imposed  upon   the  General  Equa- 
tion,    195 

Proof  that  these  exhaust  the  varieties  of  its  Locus,  .         .         .  196 

Conies  defined  :  Classification  into  Three  Species,       ....  197 
Re'sume  of  argument  for  the  theorem  :    Every  Equation  of  the  Second 

Degree  in  two  variables,  represents  a  Conic,     ....  197 


CHAPTER    SECOND. 

THE  MODERN  OEOMETRY:-TRILINEAR  AND  TANGENTIAL 
CO-ORDINATES. 

SECTION  I.  —  TRILINEAR  CO-ORDINATES. 

Trilinear  Method  of  representing  a  Point, 199 

Origin  of  the  Method:  the  Abridged  Notation,          ....  200 

Geometric  meaning  of  the  constant  k  in  a  +  &/?=  0,  .         .         .         .  201 

Interpretation  of  the  equations  a  ±  &/?  =  0,  a  ±  0  =  0,         .         .         .  201 
The  Notation  extended  to  equations  in  the  form  Ax  +  By  +  C=  0,    .  202 
Meaning  of  the  equation  la  -t-  mft  +ny  =  0  :  condition  that  it  repre- 
sent any  Right  Line,          ........  203 

Examples :  Any  line  of  Quadrilateral,  in  terms  of  any  Three,    .  206 


CONTENTS.  xi 

PAGE. 

The  symbols  a,  /?,  y  may  be  considered  as  Co-ordinates,   .         .         .  207 
Peculiar  Nature  of  Trilinear  Co-ordinates  :  each  a  Determined  Func- 
tion of  the  other  two, 208 

Equation  expressing  this  Condition  is  oa  +  b/3  +  cy  =  M,     .         .  209 
General    trilinear    symbol   for   a    Constant ;  namely,  7c  (a  sin  A  +  ~ 

/I  sin  B  +  -y  sin  C], 210 

To  render  homogeneous  any  given  equation  in  Trilinears,  .         .  210 

Trilinear  Equation  to  Right  Line, 212 

"              "                  "          "     parallel  to  a  given  one,        .         .  212 
"               "                  "           "      situated  at  infinity,      .         .         .212 
Condition,  in  Trilinears,  that  two  Right  Lines  shall  be  at  right  angles 

to  each  other, 213 

Trilinear  Equation  to  Right  Line  joining  Two  Fixed  Points,     .         .  214 

"                 "             any  Conic,  referred  to  Inscribed  Triangle,       .  215 

Circle,                 "                     "                "      .  216 

Same  for  any  Concentric  Circle,    .         .  216 

"                 "                 "       Triangle  of  Reference  having  any  sit- 
uation,    217 

General  Equation  of   Second  Degree  in  Trilinears :    *.  e.,  Trilinear 
Equation  to  any  Conic,  Triangle  of  Reference  having  any  sit- 
uation,       ...........  218 

Trilinear  Equations  to  Chord  and  Tangent  of  any  Conic,  .         .         .  219 

Examples  of  Trilinear  Notation  and  Conditions,          .         .         .  220 


SECTION  II. — TANGENTIAL  CO-ORDINATES. 

In   Tangential  system,   Lines  are  represented  by  co-ordinates,  and    • 

Points  by  equations,  ........     225 

Cartesian  Co-efficients  are  Tangential  Co-ordinates  :  Tangential  Equa- 
tions are  Cartesian  Equations  of  Condition  —  namely,  that  a 
Line  shall  pass  through  two  Consecutive  Points  on  a  given 
Curve, 225 

Geometric  interpretation  of  Tangentials :   how  they  represent  a  LOCKS. 

Reason  for  Name,      .         .         . 226 

The  Right  Line  in  the  Tangential  System,         .....     227 

ENVELOPES  denned  :  Condition  that  a  Right  Line  shall  touch  a  Curve 
is  the  Tangential  equation  to  the  Curve ;  or  simply  the  Equa- 
tion to  the  Envelope  of  the  Line,  ......  228 

Development  of  the  Tangential  Equation  to  a  Conic,  referred  to  In- 
scribed Triangle, 228 

Reciprocal    relation   between   Points   and   Lines:   the   Principle    of 

Duality, 235 

Description  of  the  Method  of  Reciprocal  Polars  :   its  relation  to  the 

Modern  Geometry,      .........     238 

Examples  illustrating  Tangentials, 242 


Xll  CONTENTS. 

PART  II.     ON  THE  PROPERTIES  OF  CONICS. 
CHAPTER    FIKST. 

THE  RIGHT  LINE.  PAQE 

Area  of  a  Triangle  in  terms  of  the  Co-ordinates  of  its  vertices,         .     246 
"        "  "          given  the  Right  Lines  which  inclose  it,      .         .     246 

Compound   ratio   of  segments   of  the    Three   Sides   by   any   Trans- 

versal =  -  1,      ..........     248 

Compound  ratio  of  segments  of  the  Three  Sides  by  any  three  Con- 

vergents  =  +1,  .         .         .         .         .         .         .         .         .     248 

Various   cases    of   Three   Convergents    occurring    in   any  Triangle, 

solved  by  the  Abridged  Notation,     ......     249 

Further   application    of    Trilinears  :     The    property   of   Homology  ; 

Axis  and  Center  of  Homology,          ......     250 

Quadrilaterals  —  when  Complete:    Centers  of  their  three  Diagonals 

lie  on  one  Right  Line,       ........     251 

Harmonic  and  Anharmonic  Properties  :......     253 

Constant  ratio  among  Segments  of  Transversals  to  any  Linear 

Pencil,          ..........     253 

Harmonic  and  Anharmonic  Pencils,     ......     254 

Anharmonic  of  a,  (3,  a  +  kfl,  a  +  k'{3  =  ^  ,  .....     255 

a,  /?,  a  +  &/?,  a—  fc/?,  form  a  Harmonic  Pencil,       ....     255 
Anharmonio   of  any    Pencil    a  +  k/3,  a  +  10,  a  +  mfi,  a  +  n/3  — 


(*-»)(*-*)'  ' 
Definition  of  Homographic  Systems  of  lines,       .         .         .         .     257 

Examples  involving  properties  of  the  Right  Line  : 

Triangles,      ...........     257 

Harmonics  of  a  Complete  Quadrilateral,      .....     258 


CHAPTER    SECOND. 

THE  CIRCLE. 
I.  THE  Axis  OP  X: 

Every  Ordinate  a  mean  proportional  between  the  correspond- 
ing segments, 259 

Every  Right  Line  meets  the  Curve  in  Two  Points,         .         .     260 
Discrimination    between   Real,   Coincident,   and    Imagi- 
nary points, 260 

Chords  defined.     Equation  to  any  Chord,        .         .         .         .262 
II.  DIAMETERS  : 

Definition :    Locus    of   middle   points    of    Parallel    Chords. 

Equation, 263 


CONTENTS.  xiii 

PAGE. 

Every  diameter  passes  through  Center,  and  is  perpendicular 

to  bisected  Chords,          .......     264 

CONJUGATE  DIAMETERS  defined  —  each  bisects  Chords  parallel 

to  the  other, 264 

Conjugates  of  the  Circle  are  at  right  angles,    .         .         .     265 

III.  TANGENT  : 

Definition :    Chord  meeting  Curve  in  Two  Coincident  Points,  265 

Equation, 266 

Condition  that  a  Right  Line  shall  touch  Circle.     Auxiliary 

Angle, 267 

Analytic  Construction  of  Tangent  through  (x',y'):   Two  Tan- 
gents, real,  coincident,  or  imaginary,  ....  268 

Length  of  Tangent  from  (x,  y)  =  !/£", 269 

Subtangent  —  its  definition  and  value,    ....      270,  271 

IV.  NORMAL: 

Definition  of  Normal.     Equation,    ......  270 

The  Normal  to  Circle  passes  through  Center.     Length  con- 
stant,        271 

Subnormal  —  its  definition  and  value, 271 

V.  SUPPLEMENTAL  CHORDS : 

Definition  :  Equation  of  Condition,  ....       271,  272 

In  the  Circle,  they  are  always  at  right  angles,       .         ,         .  272 
VI.  POLE  AND  POLAR: 

Development  of  the  conception  of  the  Polar,  ....  273 

I.  Chord  of  Contact  to  Tangents  from  (x',y'),       .         .  273 
II.  Locus  of-  intersection  of  Tangents  at  extremities  of 

convergent  chords,         ......  273 

III.  Tangent  brought  under  this  conception,    .         .         .  274 

Construction  of  Polar  from  its  definition,         ....  276 

Polar  is  perpendicular  to  Diameter  through  Pole:  —  its  dis- 
tance from  Center, 277 

Simplified  geometric  construction,     .....  277 
Distances  of  any  two  points  from  Center  are  proportional  to 

distance  of  each  from  Polar  of  the  other,       .         .         .  278 
Conjugate  and  Self-conjugate  Triangles  defined:  —  they  are 

homologous, 278,  279 

SYSTEMS  OF  CIRCLES. 
I.  SYSTEM  WITH  COMMON  RADICAL  Axis : 

Radical  Axis  defined  :   its  Equation,  S—  S'  =  0,      .         .         .280 

It  is  perpendicular  to  the  line  of  the  centers,  .         .         .      281 

Construction :  Combine  #  —  £'=-  0  with  y=  0  and  observe 

the  foregoing, 281 

The  three  Radical  Axes  belonging  to  any  three  Circles  meet 

in  one  point:    Radical  Center, 281 


xiv  CONTENTS. 

PAGE. 

To  construct  the  Radical  Axis  by  means  of  the  Radical 

Center, 281 

Radical  Axis  of  Point  and  Circle;  —  of  Two  Points,       .         .     282 
Definition  of  System  of  Circles  with  Common  Radical  Axis  ;  — 
Their   Centers   lie   on   one  Right  Line.     Their  Equa- 
tion :    x2  +  if  —  2kx  ±  ff  =  0, 282 

To  trace  the  System  from  the  equation, 283 

Locus  of  Contact  of  Tangents  from  any  point  in  the  C.  R.  A. 

is  Orthogonal  Circle, 283 

Geometric  construction  of  the  System :     Limiting  Points,    284 
Analytic  proof  of  the  existence  of  the  Limiting  Points,         .     285 
II.  Two  CIRCLES  WITH  COMMON  TANGENT  : 

To  determine  the  Chords  of  Contact, 287 

The  Tangents  intersect  on  Line  of  Centers,  and  cut  it  in  ratio 

of  Radii,         .         .         .  " 289 

Every  Right  Line  through  these  Points  of  Section  is  cut  sim- 
ilarly by  the  two  Circles 289 

The  Centers  of  Similitude, 289 

The  three  homologous  Centers  of  Similitude  belonging  to  any 
three  Circles,  lie  on  one  Right  Line.  The  Axis  of 
Similitude, 290 

THE  CIRCLE  IN  THE  ABRIDGED  NOTATION. 
If  a  Triangle  be  inscribed  in  a  Circle,  and  Perpendiculars  be  dropped 
from  any  point  in  the  Circle  upon  the  three  sides,  their  feet 

will  lie  on  one  Right  Line, 291 

Angle  between  Tangent   and  Chord  =  angle  inscribed  under  corre- 
sponding arc, 292 

Trilinear  Equation  to  the  Tangent,  referred  to  Inscribed  Triangle,  .     292 
Tangents  at  Vertices  of  Inscribed  Triangle  cut  Opposite  Sides 

in  points  lying  on  one  Right  Line,    .....     292 
Linea  joining  Vertices  of  Inscribed  Triangle   to   those  of  Tri- 
angle formed  by  Tangents  meet  in  one  Point,  .         .     292 

Radical  Axis  in  Trilinears, 293 

Examples  on  the  Circle, 293 


CHAPTER    THIRD. 

THE  ELLIPSE. 
I.  THE  CURVE  REFERRED  TO  ITS  AXES. 

THE  AXES. 

THEOREM  I.   Focal  Center  bisects  the  Axes.     Corresponding  inter- 

X2          II2 

pretation  of  —  -f  —z  =  1, 297 


CONTENTS.  xv 

PAGE. 

THEOREM      II.  Foci  fall  within  the  Curve, 298 

THEOREM    III.  Vertices  of  Curve  equidistant  from  Foci,  .         .     298 

THEOREM     IV.  Sum  of  Focal  Radii  =  length  of  Transverse  Axis.    To 

construct  Curve  by  Points,    .....     298 

THEOREM       V.  Semi-conjugate    Axis    a    geometric    mean    between 

Focal  Segments  of  Transverse,     ....     299 
Cor.  Distance  from  Focus  to  Vertex  of  Conjugate  = 

Semi-transverse.     To  construct  Foci,  .         .         .     299 

THEOREM     VI.  Squares  of  Ordinates  to  Axes   are  proportional  to 

Rectangles  of  corresponding  Segments,       .         .     300 
Cor.   The  Latus  Rectum,  and  its  value,       .         .         .     300 

THEOREM   VII.  Squares  of  Axes  are  to  each  other  as  Rectangle  of 

any  two  segments  to  Square  of  Ordinate,     .         .     301 

THEOREM  VIII.  Ordinate  of   Ellipse    :    Corresponding   Ordinate   of 
Circumscribed  Circle  : :  Semi-conjugate  :    Semi- 
transverse,    ........     301 

Cor.  I.   Analogous  relation  to  Inscribed  Circle.    Con- 
structions for  the  Curve, 302 

a2  —  I2 
Cor.  2.  Interpretation  of —  =  ez  :   e  defined  as 

the  Eccentricity, 303 

THEOREM  IX.  The  Focal  Radius  of  the  Curve  is  a  Linear  Function 

of  corresponding  Abscissa,    .....     304 

LINEAR  EQUATION  to  Ellipse  :     p  =  a  ±  ex,        .         .304 

Verification  of  the  Figure  of  the  Curve  by  means  of  its  equation,     .     304 

DIAMETERS. 

Diameters :       Equation    to    Locus    of    middle    points    of    Parallel 

Chords, 305 

THEOREM       X.  Every  Diameter  is  a  Right  Line  passing  through 

the  Center,  306 

Cor.    Every  Right  Line  through  Center  is  a  Diam- 
eter,        306 

THEOREM     XI.  Every  Diameter  of  an  Ellipse  cuts  Curve  in  Two 

Real  Points,  ....  .     306 

Length  of  Diameter,  in  the  Ellipse, 306 

THEOREM    XII.  Transverse  Axis  the  maximum,  and  Conjugate  the 

minimum  Diameter,         ......     306 

THEOREM  XIII.  Diameters  making    supplemental    angles  with  Axis 

Major  are  equal,    .......     307 

Cor.    Given  the  Curve,  to  construct  the  Axes,  .         .     307 
THEOREM  XIV.  If  a  Diameter  bisects   Chords  parallel  to  a  second, 

second  bisects  Chords  parallel  to  first,          .         .     307 

CONJUGATE  DIAMETERS  defined :    Ordinates  to  any  Diameter,    .         .     308 

To  construct  a  pair  of  Conjugates,  I  308 


xvi  CONTENTS. 

PACK. 

Equation     of     Condition     to     Conjugates,      in      the      Ellipse,      is 

tan  0.  tan  Q'  = ,    .         .         .         .  .     308 

or 

THEOREM        XV.  Conjugates  in  the  Ellipse  lie  on  opposite  sides  of 

Axis  Minor,      .......     308 

Equation  to  Diameter  Conjugate  to  that  through  any  given  point,  .     309 
Cor.   The  Axes  are  a  case  of  Conjugates,      .         .     309 
(Jiven  the  co-ordinates  to  extremity  of  any  Diameter,  to  find  those 

to  extremity  of  its  Conjugate,  ......     309 

THEOREM      XVI.  Abscissa  to  extremity  of  Diameter  :  Ordinate  to 
extremity  of  its  Conjugate  : :  Axis    Major  : 

Axis  Minor, 310 

THEOREM    XVII.  Sum  of  squares    on   Ordinates  to   extremities  of 

Conjugates,  constant  and  =  62,  .         .     310 

Length   of  any   Diameter  in  terms   of  Abscissa  to  extremity  of  its 

Conjugate, 310 

THEOREM  XVIII.  Square  on  any  Semi-diameter  =  Rectangle  Focal 

Radii  to  extremity  of  its  Conjugate,        .         .     311 
THEOREM      XIX.  Distance,  measured  on  a  Focal  Chord,  from  ex- 
tremity of  any  Diameter  to  its  Conjugate,  is 
constant,  and  equals  the  Semi-Major,     .         .     311 
THEOREM        XX.  Sum  of  squares  on  any  two  Conjugates  is  constant 

and  =  sum  squares  on  Axes,  .         .         .     312 

Angle  between  any  two  Conjugates,      .         .         .     312 
THEOREM      XXI.  Parallelogram  under  any  two  Conjugates,  constant 

and  =  Rectangular  under  Axes,         .         .         .     313 
Cor.  1.    Curve  has  but  one  set  of  Rectangular  Con- 
jugates,     313 

Cor.  2.   Inclination    of     Conjugates    is     maximum 

when  a'  =  V, 314 

THEOREM    XXII.  Equi-conjugates :    they  are  the  Diagonals  of  Cir- 
cumscribed Rectangle, 314 

Cor.     Curve    has    but    one    pair    of    Equi-conju- 
gates,         314 

Anticipation  of  the  Asymptotes  in  the  Hyperbola,     ....      315 

THE    TANGENT. 

Equation  to  the  Tangent,  referred  to  the  Axes,          ....      315 

Condition  that  a  Right  Line  touch  an  Ellipse.     Eccentric  Angle,       .      316 
Analytic   construction   of    the   Tangent    through    any  point :     Two, 

real,  coincident, or  imaginary,  ......     317 

THEOREM  XXIII.  Tangent  at  extremity  of  any  Diameter  is  parallel 

to  its  Conjugate,      ......      318 

Cor.   Tangents  at  extremities  of  any  Diameter  are 

parallel.      Circumscribed  Parallelogram,         .      318 


CONTENTS.  xvii 

PAGE. 

THEOREM      XXIV.   Tangent    bisects    the    External   Angle    between 

Focal  Radii  of  Contact,         .         .         .         .319 
Cor.  1.    To  construct  a  Tangent  to  the  Ellipse  at 

a  given  point, 319 

Cor.  2.    Derivation  of  the  term  Focus,  .         .     319 

Intercept  by  Tangent  on  Axis   Major:    Constructions  for  Tangent 

by  means  of  it, 320 

The  SUBTANGENT  defined :    Distinction   between   Subtangent  of  the 

Curve  and  of  a  Diameter,          .......    320 

THEOREM       XXV.  Subtangent  of  Curve  is  Fourth  Proportional  to 
Abscissa  of  Contact  and  the  corresponding 
segments  of  Axis  Major,       ....     321 

Cor.   Construction  of  Tangent  by  means  of  Cir- 

cumscr.  Circle  :  Subtan.  not  function  of  &,    .    321 
THEOREM      XXVI.  Perpendicular  from  Center  on  Tangent  is  Fourth 
Proportional    to    the     corresponding    Semi- 
conjugate  and  the  Semi-axes  :  p  =  —  - ,         .    322 

Length  of  the  Central  Perp'r  in  terms  of  its  angles  with  the  Axes,     .    322 

THEOREM   XXVII.  Locus   of  Intersection  of  Tangents    cutting   at 

right  angles  is  Concentric  Circle,          .         .    323 

Focal  Perpendiculars  upon  Tangent:   their  length,    ....     323 

THEOREM  XXVIII.  Focal   Perpendiculars    on   Tangent    are   propor- 
tional to  adjacent  Focal  Radii,    .         .         .    324 

THEOREM     XXIX.  Rectangle  under  Focal  Perpendiculars,  constant 

and  -  62, 324 

THEOREM        XXX.  Locus  of  foot  of  Focal  Perpendicular  is  the  Cir- 
cumscribed Circle, 324 

Cor.   Method  of  drawing  the  Tangent,  common 

to  all  Conies, 325 

THEOREM     XXXI.  If  from  any  Point  within  a  circle  a  Chord  be 
drawn,   and    a   perpendicular  to   it   at   the 
point  of  section,  the  Perpendicular  is  Tan- 
gent to  an  Ellipse,        .         .         .         .         .    326 
Cor.   The  Ellipse  as  Envelope,    .         .  .326 

THEOREM   XXXII.  Diameters  through  feet  of  Focal  Perpendiculars, 

parallel  to  Focal  Radii  of  Contact,      .         .    327 
Cor.   Diameters  parallel  to  Focal  Radii  of  Con- 
tact, meet  Tangent  at  the  feet  of  the  Focal 
Perpendiculars,    and    are    of    the    constant 
length  =  2n, 327 

THE   NORMAL. 

Equation  to  the  Normal,  referred  to  the  Axes, 328 

THEOREM  XXXIII.  Normal  bisects  the  Internal  Angle  between  Focal 

Radii  of  Contact, 328 

An.  Ge.  2. 


xviil  CONTENTS. 

PACK. 

Cor.  1.   To  construct  a  Normal  at  given  point 

on  the  Ellipse, 329 

Cor.  2.  To  construct  a  Normal  through  any 

point  on  Axis  Minor,  ....  329 

Intercept  of  Normal  on  Axis  Major :  Constructions  by  means  of  it,  329 
THEOREM  XXXIV.  Normal  cuts  distance  between  Foci  in  segments 

proportional  to  adjacent  Focal  Radii,  .  330 

The  SUBNORMAL  defined  :  Subnormal  of  the  Curve  —  its  length,  .  330 
THEOREM  XXXV.  Normal  cuts  its  Abscissa  in  constant  ratio  = 

a2-b2 

IT-' 33° 

Length  of  Normal  from  Point  of  Contact  to  either  Axis,  .         .     330 

THEOREM     XXXVI.  Rectangle  under  Segments  of  Normal  by  Axes  = 

Square  on  Conjugate  Semi-diameter,  .     331 

Cor.   Equal,   also,   to    Rectangle    under   corre- 
sponding Focal  Radii,         ....     331 
THEOREM   XXXVII.  Rectangle  under  Normal  and  Central  Perpen- 
dicular, constant  and  =  a2,         .         .         .     331 

SUPPLEMENTAL  AND  FOCAL   CHORDS. 

Equation  of  Condition  to  Supplemental  Chords,  ....  332 
THEOREM  XXXVIII.  Diameters  parallel  to  Supplemental  Chords  are 

Conjugate, 332 

Cor.  1.    To  construct  Conjugates  at  a  given  in- 
clination.    Caution, 333 

Cor.  2.   To    construct    a    Tangent    parallel    to 

given  Right  Line, 333 

Cor.  3.    To   construct   the  Axes   in   the    empty 

Curve, 333 

Focal  Chords  —  Special  Properties, 334 

THEOREM  XXXIX.  Focal  Chord  parallel  to  any  Diameter,  a  third 
proportional  to  Axis  Major  and  the  Di- 
ameter,   335 

II.   THE  CURVE  REFERRED  TO  ANY  Two  CONJUGATES. 

DIAMETRAL   PROPERTIES. 

Equation  to  Ellipse,  referred  to  Conjugates, 336 

THEOREM       XL.  Squares  on  Ordinates  to  any  Diameter,  proportional 

to  Rectangles  under  corresponding  Segments,       337 

THEOREM    XLI.  Square  on  Diameter  :  Square  on  Conjugate  : :  Rect- 
angle under  Segments  :  Square  on  Ordinate,     .     337 

THEOREM  XLII.  Ordinate   to   Ellipse    :    Corresponding   Ordinate    to 

Circle  on  Diameter  : :  V  :    a',  .         .         .     338 

Cor.  1.   Given   a  pair  of    Conjugates,  to    construct 

the  Curve,  .    338 


CONTENTS. 


xix 


Cor.  2.   General  interp'n  of  x2  +  if  =  a" :  Ellipse, 

referred  to  Equi-conjugates,          .         .         .     339 
Figure  of  the  Ellipse  with  respect  to  any  two  Conjugates,         .         .     339 

CONJUGATE    PROPERTIES    OF  THE   TANGENT. 

Equation  to  Tangent,  referred  to  Conjugates, 330 

THEOREM  XLIII.  Intercept  of  Tangent  on  any  Diameter,  third 
proportional  to  Abscissa  of  Contact  and  the 

Semi-diameter:    x=—,       .         .         .         .     340 

THEOREM  XLIV.  Rectangle  under  Intercepts  by  Variable  Tangent 
on  Two  Fixed  Parallel  Tangents,  constant 
and  =  Square  on  parallel  Semi-diameter,  .  341 

THEOREM  XLV.  Rectangle  under  Intercepts  on  Variable  Tangent 
by  Two  Fixed  Parallel  Tangents,  variable 
and  =  Square  on  Semi-diameter  parallel  to 
Variable, 341 

THEOREM      XL  VI.  Rectangle  under  Intercepts  on  Variable  Tangent 
by  any  two  Conjugates  equals   Square  on 
Semi-diameter  parallel  to  Tangent,     .         .     342 
Cor.  1.   Diameters  through  intersections  of  Va- 
riable  Tangent    with   Two   Fixed  Parallel 

ones,  are  Conjugate, 342 

Cor.  2.   Given  two    Conjugates  in  position  and 
magnitude,  construct  the  Axes,    . 


THE  SUBTANGENT  TO  ANY  DIAMETER  :    its  length  = 


342 
343 


X 

Cor.    Construction    of    Tangent    by   means    of 

Auxiliary  Circle,  ......     343 

THEOREM    XLVII.  Rectangle   under  Subtangent   and  Abscissa   of 

Contact  :  Square  on  Ordinate  :  :  a'2  :  I'2,      .     344 
THEOREM  XLVIII.  Tangents  at  extremities  of  any  Chord  meet  on 

its  bisecting  Diameter,  ....     345 

PARAMETERS. 

Parameter  to  any  Diameter  defined :    Third  proportional  to  Diam- 
eter and  Conjugate, 345 

Parameter  of  the  Curve  :    identical  with  Latus  Rectum,     .         .         .     345 
THEOREM      XLIX.  In  the  Ellipse,  no  Parameter  except  the  Princi- 
pal is  equal  in  value  to  the  Focal  Double 
Ordinate, 346 

THE    POLE    AND    THE    POLAR. 

Development  of  the  Equation  to  the  Polar  :    Definition,   .         .    346—349 
Cor.    Construction  of  Pole    or  Polar    from  its 

definition, 349 


xx  CONTENTS. 

PAGE. 

THEOREM        L.  Polar  of  any  Point,  parallel  to  Diameter  Conjugate 

to  the  Point, 350 

Special  Properties  :    Polar  of  Center ;  —  of  any  point  on  axis  of  x ;  — 
on  Axis  Major,  ......... 

Cor.  Second   geometric  construction  for  Polar, 
POLAR  OF  Focus  :   its  distance  from  center,  and  its  direction,  . 
THEOREM      LI.  Ratio  between  Focal   and  Polar  distances  of  any 

point  on  Ellipse,  constant  and  =  e,       .  .     352 

Cor.  1.   On  the  construction  of  the  Ellipse  according 

to  this  theorem,  .         .         .         .         .         .352 

Cor.  2.   Polar  of  Focus  hence  called  the  Directrix,    .    353 
Cor.  3.   Second  basis  for  the  name  Ell-ipxe,        .         .    353 
THEOREM  LII.  Line  from  Focus  to  Pole  of  any  Chord,  bisects  focal 

angle  which  the  Chord  subtends,         .         .         .    354 
Cor.   Line  from  Focus  to  Pole  of  Focal  Chord,  per- 
pendicular to  Chord, 354 

III.    THE  CURVE  REFERRED  TO  ITS  Foci. 

Interpretation  of  the  Polar  Equations  to  the  Ellipse,         .         .         .  355 

Development  of  the  Polar  Equation  to  a  Tangent,    .  *      .         .         .  355 
Polar  proof  of  Theorem  XIX  compared  with  former  proof,  and  with 

that  by  pure  Geometry,     ........  356 

IV.   AREA  OF  THE  ELLIPSE. 
THEOREM  LIII.  Area  of  Ellipse  =  T  times   the  Rectangle  under  its 

Semi-axes,  . 358 

V.    EXAMPLES  ON  THE  ELLIPSE. 
Loci,  Transformations,  and  Properties,       ......    358 


CHAPTER    FOURTH. 

THE  HYPERBOLA. 

I.    THE  CURVE  REFERRED  TO  ITS  AXES. 
THE  AXES. 

THEOREM      I.  Focal  Center  bisects  the  Axes.     Corresponding  inter- 

x2      if 

pretation  of  -  —  J-  =•=  1, 363 

a        b 

The  Axis  Conjugate,  conventional :    Equation  to  the  Conjugate  Hy- 
perbola,        364 

THEOREM    II.  Foci  full  without  the  Curve, 365 

THEOREM  III.  Vertices  of  Curve,  equidistant  from  Foci,    .         .         .  365 
THEOREM  IV.  Difference  Focal  Radii  =  Transverse.     The  Curve  by 

Points, 365 


CONTENTS. 


xxi 


THEOREM       V.  Conventional  Semi-conjugate,  geometric  mean   be- 
tween Focal  Segments  of  Transverse, 
Cor.  Dist.  from  Center  to  Focus  =  dist.  between  ex- 
tremities of  Axes.     To  construct  Foci, 

THEOREM     VI.  Squares  on  Ordinates  to  Axes,  proportional  to  Rect- 
angles under  corresponding  Segments, 
Cor.   The  Latus  Rectum,  and  its  value, 
THEOREM   VII.  Squares  on  Axes  are  as  Rectangle  under  any  two 

Segments  to  Square  on  their  Ordinate, 

Analogy  of  Hyperbola  to  Ellipse,  with  respect  to  Circle  on  Trans- 
verse, defective.    Circle  replaced  by  the  Equilateral  Hyperbola, 
THEOREM  VIII.  Ordinate  Hyperbola  :  Corresponding  Ordinate  of  its 
Equilateral  : :  b  '  a, 


Cor.  Interpretation  of — 


?z.     e  defined  as  the 


Eccentricity,          ....... 

THEOREM     IX.  Focal  Radius  of  Curve,  a  Linear  Function  of  corre- 
sponding Abscissa,     ...... 

LINEAR  EQUATION  to  Hyperbola:  p  =  ex±a,  . 

Verification  of  Figure  of  Curve  by  means  of  its    equation, 

DIAMETERS. 


366 
366 

367 
367 

367 


368 


369 

370 
371 
371 


Diameters  :   Equation  to  Locus  of  middle  points  of  Parallel  Chords,       371 
THEOREM        X.  Every  Diameter  a  Right  Line  passing  through  the 

Center, .371 

Cor.   Every  Right  Line   through  center  is  a  Diam- 
eter,     372 

THEOREM     XI.  "Every  Diameter  cuts  Curve  in  Two  Real  Points" 

untrue  for  Hyperbola, 372 

Cor.  1.  Limit  of  those  diameters  having  real  intersec- 


tions: 


372 


Cor.  2.   All  diameters  cutting  Hyperbola  in  Imagi- 
nary Points,  cut  its  Conjugate  in  Two  Real  ones,    373 

Length  of  Diameter,  in  the  Hyperbola, 

THEOREM  XII.  Each  Axis  the  minimum  diameter  for  its  own  curve,  . 
THEOREM  XIII.  Diameters  making  supplemental  angles  with  Trans- 
verse are  equal,  ...... 

Cor.  Given  the  Curve,  to  construct  the  Axes,    . 
THEOREM  XIV.  If  a  Diameter  bisects  Chords  parallel  to  a  second, 

second  bisects  those  parallel  to  first, 
CONJUGATE  DIAMETERS  :     Ordinates,    .         ... 

To  construct  a  pair  of  Conjugates,   .... 

Equation  of  Condition  to  Conjugates,  in  Hyperbola  :  tan  0.  tan  9'  —  — -2 ,    375 


374 

374 

374 

374 

374 


xxii  CONTENTS. 

PAGE. 

THEOREM         XV.  Conjugates  in  the  Hyperbola  lie  on  same  side  of 

Conjugate  Axis, 375 

Equation  to  Diameter  Conjugate  to  that  through  any  given  point,    .     375 
Cor,   The  Axes  are  a  case  of  Conjugates,       .         .     376 
Given  co-ordinates   to   extremity  of  Diameter,  to  find  those  to  ex- 
tremity of  its  Conjugate, 376 

THEOREM      XVI.  Abscissa  ext'y  of  any  Diameter  :  Ordinate  ext'y 

of  its   Conjugate   : :    Transverse  :   Conjugate,     376 
THEOREM    XVII.    Diff.  squares  on  Ordinates  to  extremities  of  Con- 
jugates, constant  and  =  b2,  .         .         .     376 
Length   of  any   Diameter  in  terms  of  Abscissa  to  extremity  of  its 

Conjugate, 377 

THEOREM  XVIII.  Square  on  any  Semi-diameter  =  Kectangle  Focal 

Kadii  to  extremity  of  its  Conjugate,       .         .     378 
THEOREM      XIX.  Distance,  measured  on  Focal  Chord,  from  extrem- 
ity of  any  Diameter  to  its  Conjugate,  constant 
and  equal  to  Semi-Transverse,          .         .         .     378 
THEOREM        XX.  Difference  of  squares  on  any  two  Conjugates,  con- 
stant and  =  difference  of  squares  on  Axes,        378 
Angle  between  any  two  Conjugates,      .         .         .    379 
THEOREM      XXI.  Parallelogram  under  any  two  Conjugates,  constant 

and  =  Rectangular  under  Axes,         .         .         .     379 
Cor.  1.    Curve    has    but    one    set    of    Rectangular 

Conjugates, 380 

Cor.  2.    Inclination  of  Conjugates  diminishes  with- 
out limit :   the  conception  of  Equi-conjugates 
replaced  by  that  of  Self -conjugates,          .         .     380 
THEOREM    XXII.  The  Self-conjugates  in  the  Hyperbola  are  Diago- 
nals of  the  Inscribed  Rectangle,     .         .         .     381 
Cor.  Curve  has   two,  and  but  two,  Self-conjugates,     381 
Analogy  of  the  Self-conjugates  to  the  Equi-conjugates  of  the  Ellipse,     382 

THE   TANGENT. 

Equation  to  Tangent,  referred  to  the  Axes,  .....  382 
Condition  that  a  Right  Line  touch  an  Hyperbola.  Eccentric  Angle,  .  383 
Analytic  construction  of  the  Tangent  through  any  point :  Two,  real, 

coincident,  or  imaginary, 384 

THEOREM  XXIII.  Tangent  at  extremity  of  any  Diameter,  parallel 

to  its  Conjugate, 385 

Cor.    Tangents  at  extremities  of  any  Diameter  are 

parallel.     To  circumscribe  Parallelogram,     .     385 
THEOREM  XXIV.   Tangent  bisects  the  Internal  Angle  between  Focal 

Radii  of  Contact, 386 

Cor.  1.    To  construct  Tangent  to  Hyperbola,  at  a 

given  point, 386 

Cor.  2.   Derivation  of  the  term  Focus,  .         .         .386 


CONTENTS. 


XXlll 


PAGE. 

Intercept  by  Tangent  on  the   Transverse  Axis :    Constructions  by 

means  of  it, 387 

The  SUBTANGENT  to  the  Hyperbola 387 

THEOREM  XXV.  Subtangent  of  Curve  a  Fourth  Proportional  to 
Abscissa  of  Contact  and  corresponding  seg- 
ments of  Transverse, 387 

Cor.  1.  Defect  supplied  in  the   analogy  between 
Hyperbola   and    Ellipse,   respecting   Circle 

on  2a, 388 

Cor.  2.  Construction  of  Tangent  by  means   of 

Inscribed  Circle, 388 

THEOREM  XXVI.  Central  Perpendicular  on  Tangent,  a  Fourth  Pro- 
portional to  corresponding  Semi-conjugate 

and  the  Semi-axes  :  p  =  — ,        .         .         .     389 
o 

Length  of  Central  Perpendicular  in  terms  of  its  angles  with  Axes,  .     389 

THEOREM    XXVII.  Locus  of  Intersection  of  Tangents  cutting   at 

right  angles  is  Concentric  Circle,        .         .     390 

Focal  Perpendiculars  on  Tangent:    their  length,         ....     390 

THEOREM  XXVIII.  Focal  Perpendiculars  proportional  to  adjacent 

Focal  Radii, 390 

THEOREM      XXIX.  Rectangle  under  Focal  Perpendiculars,  constant 

and  =  W,       .  ...     391 

THEOREM  XXX.  Locus  of  foot  of  Focal  Perpendicular  is  In- 
scribed Circle, 391 

Cor.   To  draw  Tangent  by  the  method  common 

to  all  Conies, 391 

THEOREM      XXXI.  If,  from  any  point  without  a  Circle,  a  Chord  be 
drawn,  and  a  perpendicular  to   it   at   the 
point  of  section,  the  Perpendicular  is  Tan- 
gent to  an  Hyperbola,  ....     392 
Cor.   The  Hyperbola  as  Envelope,       .         .         .     392 

THEOREM    XXXII.  Diameters  through  feet  of  Focal  Perpendiculars 

are  parallel  to  Focal  Radii  of  Contact,         .     393 
Cor.  Diameters  parallel  to  Focal  Radii  of  Con- 
tact, meet   Tangent   at   the   feet   of   Focal 
Perpendiculars,    and    are   of   the    constant 
length  =  2a, 393 

THE    NORMAL. 

Equation  to  the  Normal,  referred  to  the  Axes, 393 

THEOREM  XXXIII.  Normal    bisects    the    External    Angle    between 

Focal  Radii  of  Contact,         .         .         .         .394 
Cor.  1.   If  Ellipse  and  Hyperbola  are  confocal, 
Normal  to  one  is  Tangent  to  other  at  inter- 
section,         .......    394 


xxiv  CONTENTS. 

PAGE. 

Cor.  2.   To  construct  Normal  at  any  point  on 

Hyperbola, 394 

Cor.  3.    To    construct    Normal    through    any 

point  on  Conjugate  Axis,  .         .         .    394 

Intercept  of  Normal  on  Transverse  Axis :    Constructions  by  means 

of  it, 395 

THEOREM      XXXIV.  Normal  cuts  distance  (produced)  between  Foci 

in  segments  proport'l  to  Focal  Radii,       .    395 

The  SUBNORMAL  :    Subnormal  of  the  Hyperbola — its  length,     .         .    396 
THEOREM        XXXV.  Normal    cuts    its   Abscissa    in    the    constant 

ratio  =  , 396 

Length  of  Normal  from  Point  of  Contact  to  either  Axis,           .         .    396 
THEOREM      XXXVI.  Rectangle    under    Segments    of    Normal    by 
Axes  =  Square  on  Conjugate  Semi-diam- 
eter,   396 

Cor.   Equal,  also,   to   Rectangle  under  corre- 
sponding Focal  Radii,        ....    396 
THEOREM    XXXVII.  Rectangle  under  Normal  and  Central  Perp'r 

on  Tangent,  constant  and  =  a2,         .         .    397 

SUPPLEMENTAL    AXD    FOCAL    CHORDS. 

Equation  of  Condition  to  Supplemental  Chords,         ....    397 
THEOREM  XXXVIII.  Diameters  parallel  to  Supplemental  Chords  are 

Conjugate, 397 

Cor.  1.   To    construct   Conjugates  at  a   given 

inclination, 397 

Cor.  2.   To    construct   a   Tangent   parallel   to 

given  Right  Line, 398 

Cor.  3.    To  construct  the  Axes  in  the  empty 

Curve, 398 

Focal  Chords — Properties  analogous  to  those  for  the  Ellipse,    .         .    398 
THEOREM      XXXIX.  Focal  Chord  parallel  to  any  Diameter,  a  third 
proportional    to  the  Transverse    and  the 
Diameter, 399 

II.  THE  CURVE  REFERRED  TO  ANY  Two  CONJUGATES. 

DIAMETRAL    PROPERTIES. 

Equation  to  Hyperbola,  referred  to  Conjugates.  Conjugate  and 

Equilateral  Hyperbola,  ........  400 

THEOREM  XL.  Squares  on  Ordinates  to  any  Diameter,  proportional 

to  Rectangles  under  corresponding  Segments,  .  401 

THEOREM  XLI.  Square  on  Diameter  :  Square  on  its  Conjugate  : : 
Rectangle  under  Segments  :  Square  on  their 
Ordinate,  ...  .401 


CONTENTS.  xxv 

PAGE. 

THEOREM       XLII.  Ordinate  to  Hyperbola  :  Corresponding  Ordinate 

to  Equilateral  on  Axis  of  x    :  :  bf  :  a',         .    401 
Bern.  Failure  of  analogy  to  Ellipse  in  respect  to 

Diametral  Circle, 401 

Figure  of  the  Hyperbola,  with  respect  to  any  pair  of  Conjugates,      .    402 

CONJUGATE    PROPERTIES    OF  TANGENT. 

Equation  to  Tangent,  referred  to  Conjugates, 402 

THEOREM  XLIII.  Intercept  of  Tangent  on  any  Diameter,  third  pro- 
portional to  Abscissa  of  Contact  and  the 

Senai-diameter :   x  —  — - ,  .         .         .    402 

THEOREM  XLIV.  Rectangle  under  Intercepts  by  Variable  Tangent 
on  Two  Fixed  Parallel  Tangents,  constant 
and  —  Square  on  parallel  Semi-diameter,  .  403 

THEOREM  XLV.  Rectangle  under  intercepts  on  Variable  Tangent 
by  Two  Fixed  Parallel  Tangents,  variable 
and  =  Square  on  Semi-diameter  parallel  to 
Variable, •  403 

THEOREM  XLVI.  Rectangle  under  Intercepts  on  Variable  Tangent 
by  any  two  Conjugates  =  Square  on  Semi- 
diameter  parallel  to  Tangent,  .  .  .  403 
Cor.  1.  Diameters  through  intersection  of  Vari- 
able Tangent  with  Two  Fixed  Parallel  Tan- 
gents are  Conjugate, 403 

Cor.  2.  Given   two  Conjugates  in   position    and 

magnitude,  to  construct  Axes,      .         .         .    403 

xn  —  an 
THE  SUBTANGENT  TO  ANY  DIAMETER  :   its  length  = — ,      .         .    404 

Cor.  Construction  of  Tangent  by  means  of  Aux- 
iliary Circle, 404 

THEOREM  XLVII.  Rectangle  under  Subtangent  and  Abscissa  of 

Contact  :  Square  on  Ordinate  :  :  a'2  :  b'z,  .  405 

THEOREM  XL  VIII.  Tangents  at  extremities  of  any  Chord  meet  on  its 

bisecting  Diameter, 405 

PARAMETERS. 

Definitions.     Parameter  of  Hyperbola  identical  with  its  Latus  Rectum,    406 
THEOREM      XLIX.  In    the    Hyperbola,    no    Parameter  except   the 
Principal    equal    in    value    to     the    Focal 
Double  Ordinate, 406 

THE   POLE   AND    THE    POLAR. 

Development  of  the  Equation  to  the  Polar :    Definition,     .         .  406 — 408 
Cor.   Construction    of    Pole    or    Polar   from    its 

definition, 409 

An.  Ge.  3. 


XXVI 


CONTENTS. 


THEOREM  L.  Polar  of  any  Point,  parallel  to  Diameter  Conjugate 

to  the  Point, 409 

Polar  of  Center; — of  any  point  on  Axis  of  x  ; — on  Transverse  Axis,  410 

Cor.   Second  geometric  construction  for  Polar,      .  410 

POLAR  OP  Focus  :  its  distance  from  center,  and  its  direction,     .         .  410 
THEOREM         LI.  Ratio  between  Focal  and  Polar  distances  of  any 

point  on  Hyperbola,  constant  and  =  e,    .         .  411 

Cor.  1.    Curve  described  by  continuous  Motion,    .  411 

Cor.  2.    Polar  of  Focus  hence  called  the  Directrix,  412 

Cor.  3.    Second  basis  for  name  Hyperbola,     .         .412 

THEOREM       LII.  Line   from   Focus   to   Pole   of   any  Chord,  bisects 

focal  angle  which  the  Chord  subtends,    .         .  413 
Cor.   Line  from  Focus  to  Pole  of  Focal  Chord,  per- 
pendicular to  Chord, 413 


III.  THE  CURVE  REFERRED  TO  ITS  Foci. 

Interpretation  of  the  Polar  Equations  to  the  Hyperbola,  .         .         .  413 

Development  of  the  Polar  Equation  to  a  Tangent,     ....  414 

IV.    THE  CURVE  REFERRED  TO  ITS  ASYMPTOTES. 

ASYMPTOTES  denned:    Derivation  of  the  name,          ....  414 

THEOREM     LIII.  Self-conjugates  of  Hyperbola  are  Asymptotes,       .  416 

Angle  between  the  Asymptotes;  —  its  value  in  the  Eq.  Hyperb.,      .  416 

Equations  to  the  Asymptotes,     ........  416 

THEOREM      LIV.  Asymptotes  par.  to  Diag'ls  of  Semi-conjugates,     .  417 

THEOREM        LV.  Asymptotes  limits  of  Tangents,      ....  418 

THEOREM      LVI.  Perpendicular  from  Focus  on  Asymptote  =  Con- 
jugate Semi-axis,    ......  419 

THEOREM    LVII.  Focal  distance  of  any  point  on  Hyperbola  =  dis- 
tance to  Directrix  on  parallel  to  Asymptote,  .  419 
Equation  to  Hyperbola,  referred  to  its  Asymptotes,  ....  420 

Equation  to  Conjugate  Hyperbola,         .         .         .  420 
THEOREM  LVIII.  Parallelogram    under    Asymptotic    Co-ordinates, 

constant  and  =  -Q->  .....  421 

THEOREM      LIX.  Right  Lines  joining  two  Fixed  Points  on  Curve  to 
a  Variable  one,  make  a  constant  intercept  on 

Asymptote, 422 

Equation  to  the  Tangent,  referred  to  Asymptotes,      ....  422 

"  Diameter  through  any  given  point,         ....  422 

"  "          Conjugate  to  x'y' .     Equations  to  the  Axes,      .  422 

Co-ordinates  of  extremity  of  Diameter  conjugate  to  x'y' ,  .         .  423 

THEOREM         LX.  Segment  of  Tangent  by  Asymptotes,  bisected  at 

Contact, 423 

Cor.  The  Segment  =  Semi-diameter  conjugate  to 

point  of  contact, 423 


CONTENTS.  xxvn 

PAGE. 

THEOREM      LXI.  Rectangle  intercepts  by  Tangent  on  Asymptotes, 

constant  and  =  a2  +  b2, 423 

THEOREM  LXII.  Triangle  included  between  Tangent  and  Asymp- 
totes, constant  and  =  ab,  ....  424 

THEOREM  LXIII.  Tangents   at   extremities   of  Conjugates    meet  on 

Asymptotes,      .......     424 

THEOREM  LXIV.  Asymptotes  bisect  the  Ordinates  to  any  Diam- 
eter,   425 

Cor.  I.  Intercepts   on  any   Chord   between   Curve 

and  Asymptotes  are  equal,       ....    425 
Cor.  2.  Given  Asymptotes  and  Point,  to  construct 

the  Curve, 425 

THEOREM     LXV.  Rectangle  under  Segments  of  Parallel  Chords  by 

Curve  and  Asymptote,  constant  and  =  b'2,      .    426 

V.   AREA  OP  THE  HYPERBOLA. 

THEOREM  LXVI.  Area  of  Hyperbolic  Segment  equals  log.  Abscissa 
extreme  point,  in  system  whose  modulus  = 
sin  0  :  or,  A  =  sin  0.  te', 428 

VI.  EXAMPLES  ON  THE  HYPERBOLA. 
Loci,  Transformations,  and  Properties, 428 


CHAPTER    FIFTH. 

THE  PARABOLA. 
I.   THE  CURVE  REFERRED  TO  .ITS  Axis  AND  VERTEX. 

THE  AXIS. 

THEOREM      I.  Vertex    of   Curve    bisects    distance    from    Focus    to 

Directrix, 431 

Interp'n  of  symbol  p  in  yz=  4/>(at  — />),      .         .         .  431 

THEOREM    II.  Focus  falls  within  the  Curve, 431 

Transformation  to  ?/2  =  4px,          .....  432 

Cor.   To  construct  the  Curve  by  points,  .         .  432 

THEOREM  III.  Square  on  any  Ordinate  — -  Rectangle  under  Abscissa 

and  four  times  Focal  distance  of  Vertex,     .         .  432 

Cor.  Squares  on  Ordinates  vary  as  the  Abscissas,       .  432 

The  Latits  Rectum  defined.     Its  value  =  4p, 433 

Relation  between  the  Parabola  and  the  Ellipse :    proof  that  Ellipse 

becomes  Parabola  when  a  increases  without  limit,  .         .         .  433 
Cor.  1.    Analogue,  in     Parabola,  of    Circumscribed 

Circle  in  Ellipse, 434 

Cor.  2.   Interpretation  of  I )          =  l=ez:  e  de- 

V      a-     Ja  =  00 
fined  as  the  Eccentricity, 435 


xxviii  CONTENTS. 

PAGE. 

THEOREM      IV.  Focal  Radius  of  Curve,  a  linear  function  of  corre- 
sponding Abscissa,      ......  436 

LINEAR  EQUATION  to  Parabola:   p=p  +  x,     .         .  436 

Verification  of  Figure  of  Curve  by  means  of  its  equation,  .         .  437 

Nature  of  its  infinite  branch  as  distinguished  from 

that  of  Hyperbola, 437 

DIAMETERS. 

Diameters:    Equation  to  Locus  of  middle  points  Parallel  Chords,      .  438 
THEOREM        V.  Every  Diameter  is  a  Right  Line  parallel  to  the  Axis,  439 
Cor.  1.   All  Diameters  are  parallel,            .         .         .  439 
Cor.  2.   Every  Right  Line  parallel  to  Axis,  i.  <?.,  per- 
pendicular to  Directrix,  is  Diameter,          .         .  439 
THEOREM      VI.  Every  Diameter  meets  Curve  in  Two  Points  —  one 

finite,  the  other  at  infinity,         .         .         .         .440 
CONJUGATE  DIAMETERS  —  in  case  of  Parabola,  vanish  in  the  paral- 
lelism of  all  Diameters, 440 

THE    TANGENT. 

Equation  to  the  Tangent  referred  to  Axis  and  Vertex,      .         .         .    441 
Condition  that  a  Right  Line  touch  Parabola:   y  =  mx+  —  r       .         .    441 

Analytic  construction  of  Tangent  through  x'y' ' :    Two,  real,  coinci- 
dent, or  imaginary,    .........  441 

THEOREM    VII.  Tangent  at  extremity  of  any  Diameter  is  parallel  to 

its  Ordinates, 442 

Cor.   Vertical  Tangent  is  the  Axis  of  y,          .         .  443 
THEOREM  VIII.  Tangent  bisects   the   Internal  Angle    of  Diameter 

and  Focal  Radius  to  its  Vertex,         .         .         .  443 
Cor.  1.    To  construct  Tangent  at  any  point  on  Curve,  443 
Cor.  2.   Derivation  of  Term  Focus>    ....  443 
Intercept  by  Tangent  on  Axis  :    its  length  ^  #' ,     .     .         .         .         .  444 
THEOREM      IX.  Foot  of  Tangent  and  Point  of  Contact  equally  dis- 
tant from  Focus, 444 

Cor.  To  construct  Tangent  at  any  point  on  Curve,  or 

from  any  on  Axis, 444 

SUBTANGENT  TO  THE  CURVE  :   its  length  =  2x' , 445 

THEOREM        X.  Subtangent  to  Curve  is  bisected  in  Vertex,      .         .  445 
Cor.  1.   To  construct  Tangent  at  any  point  on  Curve, 

or  from  any  on  Axis, 445 

Cor.  2.   Envelope  of  lines  in  Isosceles  Triangle,       .  446 

Focal  Perpendicular  on  Tangent :    to  determine  its  length,        .         .  446 
THEOREM      XI.   Focal  perpendicular  varies  in  subduplicate  ratio  to 

Focal  distance  of  Contact,           ....  447 

Length  of  Focal  Perpendicular  in  terms  of  its  angle  with  Axis,        .  447 


CONTENTS.  xxix 

PAGE. 

THEOREM    XII.  Locus  of  foot  of  Focal  Perp'r  is  the  Vertical  Tangent,    447 
Cor.  1.    Construction  of  Tangent  by  general  Conic 

Method, 448 

Cor.  2.   Circle  to  radius  infinity  is  the  Right  Line,       448 
THEOREM  XIII.  If  from  any  Point  a  right  line    be  drawn  to  a  fixed 
Right  Line,  and  a  perpendicular  to  it  through 
the    point    of   section,    the   Perpendicular  will 

touch  a  Parabola, 449 

Cor.   The  Parabola  as  Envelope,       .         .         .         .449 
THEOREM  XIV.  Locus  of  intersection  of  Tangent  with  Focal  Chord 
a,t  any  fixed  angle  is  Tangent  of  same  inclina- 
tion to  Axis, 450 

THEOREM     XV.  The  angle  between  any  two   Tangents  to  a  Para- 
bola =  half  the  focal  angle  subtended  by  their 

Chord  of  Contact, 451 

THEOREM  XVI.  Locus  of  intersection  of  Tangents  cutting  at  right 

angles  is  the  Directrix, 451 

Cor.   New  illustration  of  Right  Line  as  Circle  with 

infinite  radius, 451 

THE    NORMAL. 

Equation  to  the  Normal,  referred  to  Axis  and  Vertex,       .         .         .  451 
THEOREM    XVII.  Normal  bisects  External  Angle  of  Diameter  and 

Focal  Radius  to  its  Vertex,     .        .         .         .452 

Cor.   To  construct  Normal  at  any  point,        .         .  452 
Intercept  by  Normal  on  Axis :   its  length  in  terms  of  the  Abscissa 

of  Contact, 452 

Constructions  for  the  Normal  by  means  of  its  Intercept,     .         .         .  453 
THEOREM  XVIII.  Foot  of  Normal  equidistant  from  Focus  with  Foot 

of  Tangent  and  Point  of  Contact,           .         .  453 

Cor.  Corresp'g  constructions  for  Normal  or  Tang.,  453 

SUBNORMAL  TO  THE  PARABOLA,    ........  453 

THEOREM      XIX.  Subnormal  to  the  Parabola,  constant  and  =  2p,    .  453 

Length  of  the  Normal  determined, 453 

THEOREM        XX.  Normal  double  corresp'g  Focal  Perpendicular,     .  454 

II.   THE  CURVE  IN   TERMS  OP  ANY  DIAMETER. 

DIAMETRAL    PROPERTIES. 

Equation  to  Parabola,  referred  to  any  Diameter  and  Vert'l  Tangent,      454 
THEOREM     XXI.  Vertex  of  any  Diameter  bisects  distance  between 
Directrix  and  the  point  in  which  the  Diam- 
eter is  cut  by  its  Focal  Ordinate,    .         .         .    455 
THEOREM  XXII.  Focal  distance  Vertex  of  any  Diameter  =  Focal 
distance    Principal    Vertex    divided    by    the 
square  of  the  Sine  of  Angle  between  Diam- 
eter and  its  Vertical  Tangent,         .         .         .456 


XXX 


CONTENTS. 


THEOREM   XXIII.  Square  on  Ordinate  to  any  Diameter  =  Rectangle 
under  Abscissa  and  four  times  Focal  distance 

of  its  Vertex, 

Cor.   Squares  on  Ordinates  to  any  Diameter  vary 
as  the  corresponding  Abscissas,     . 

THEOREM     XXIV.  Focal  Bi-ordinate  to  any  Diameter  =  four  times 

Focal  distance  of  its  Vertex, 

Rem.   Analogy  of  this   Double   Ordinate   to   the 
Latus  Rectum  peculiar  to  Parabola, 

Figure  of  the  Curve  with  reference  to  any  Diameter, 


456 


456 


457 


457 
457 


GENERAL,  DIAMETRAL  PROPERTIES  OF  THE  TANGENT. 

Equation  to  Tangent,  referred  to  any  Diameter  and  Vert'l  Tangent,  457 

Intercept  by  Tangent  on  any  Diameter:    its  length  =xr,    .       .         .  458 
THEOREM       XXV.  Subtangent   to   any  Diameter  is   bisected   in   its 

Vertex, 458 

Cor.  1.    To   construct   a   Tangent  to  a  Parabola 

from  any  point  whatever,       ....  458 

Cor.  2.   To  construct  an  Ordinate  to  any  Diameter,  458 
THEOREM    XXVI.   Tangents  at  extremities  of  any  Chord  meet  on  its 

bisecting  Diameter, 459 

THE    POLE    AND    THE    POLAR. 

Development  of  the  Equation  to  the  Polar,         ....    459—461 
Cor.   Construction  of  Pole  and  Polar  from  their 

definitions, 461 

THEOREM  XXVII.  Polar  of  any  Point,  parallel  to  Ordinates  of  corre- 
sponding Diameter, 462 

Polar  of  any  point  on  Axis  of  x;  —  on  principal  Axis,     .         .  462 

POLAR  OF  Focus  :    its  identity  with  the  Directrix,      ....    462 
THEOREM  XXVIII.  Ratio  between  Focal  and  Polar  distances  of  any 

point  on  Parabola,  constant  and  =  e,  .         .    463 
Rem.  1.    Vindication  of  original  definition  and 

construction  of  Curve,  ....    463 

Rem.  2.    Second  basis  for  the  name  Parabola,      .    464 
THEOREM      XXIX.  Line  from  Focus  to  Pole  of  any  Chord  bisects 

focal  angle  which  Chord  subtends,        .         .    464 
Cor.   Line  from  Focus  to  Pole  of  Focal  Chord 

perpendicular  to  Chord,         ....    464 
Examples :    Intercept  on  Axis  between  any  two  Polars  =  that  be- 
tween perp'rs  from  their  Poles,          ....    465 
Circle   about   Triangle    of  any   three    Tangents   passes 

through  Focus,          .......    465 


PARAMETERS. 

Parameter  defined  as  Third  Prop'l  to  Abscissa  and  its  Ordinate, 


465 


CONTENTS.  xxxi 

PAGK. 

THEOREM        XXX.  Parameter  of  any  Diameter  =  four  times  Focal 

distance  of  its  Vertex,  ....     466 

Cor.   Parameter  of  the  Curve  =  four  times  Focal 

distance  of  the  Vertex,         ....    466 

Rein.  New  interpretation  of  various   Theorems 

and  Equations,     ......    466 

THEOREM       XXXI.  Parameter   of   any    Diameter   =   its    Focal   Bi- 

ordinate, 466 

Cor.   Parameter  of  Curve  —  Latus  Rectum,          .    466 
Eem.   The  Theorem  holds  in  the  Parabola  alone 

of  all  the  Conies,  .         .         .         .         .    466 

Parameter  of  any  Diameter  in  terms  of  Abscissa  of  its  Vertex,         .    467 
"  "  "  Principal,     .         .         .         .467 

THEOREM    XXXII.  Parameter  inversely  proportional  to  sin2  of  Ver- 
tical Tangency,     ......    467 

III.    THE  CURVE  REFERRED  TO  ITS  Focus. 

Interpretation  of  the  Polar  Equation  to  Parabola,     ....    467 
Development  of  the  Polar  Equation  to  Tangent,  .         .         .     468 

IV.   AREA  OP  THE  PARABOLA. 
THEOREM  XXXIII.  Area  of  any  Parabolic   Segment  =  Two-thirds 

the  Circumscribing  Rectangle,     .         .         .    470 

V.  EXAMPLES  ON  THE  PARABOLA. 
Loci,  Transformations,  and  Properties, 470 


CHAPTEE    SIXTH. 

THE  CONIC  IN  GENERAL. 

I.  THE  THREE  CURVES  AS  SECTIONS  OF  THE  CONE. 
Definitions, 473 

Conditions  of  the  several  Sections,  and  their  Geometric  order,  .    474 

II.  VARIOUS  FORMS  OF  EQUATION  TO  THE  CONIC  IN  GENERAL. 
General  Equation  in  Rectangular  Co-ordinates  at  the  Vertex,    .         .  475 
Equation  to  the  Conic,  in  terms  of  the  Focus  and  its  Polar,       .         .  478 

Linear  Equation  to  the  Conic, 479 

Equation  to  the  Conic,  referred  to  any  two  Tangents,        .         .         .  480 

The  Conic  as  Locus  of  the  Second  Order  in  General,          .         .         .  482 

III.  THE  CURVES  IN  SYSTEM  AS  SUCCESSIVE  PHASES  OF  ONE 

FORMAL  LAW. 

Order  of  the  Curves,  as  given  by  Analytic  Conditions,      .         .         .    483 
Classification  of  the  Conies, 485 


xxxii  CONTENTS. 

PAGE. 

IV.  DISCUSSION  OP  THE  PROPERTIES  OP  THE  CONIC  IN  GENERAL. 

The  Polar  Relation, 486 

Diameters  :    Development  of  the  Center, 495 

Development  of  the  conception  of  Conjugates  and  of  the  Axes,         .  498 

Development  of  the  Asymptotes  in  general  symbols,           .         .         .  501 

Similar  Conies  defined,         .........  506 

V.  THE  CONIC  IN  THE  ABRIDGED  NOTATION. 

Fundamental  Anharmonic  Property  of  Conies,           ....  507 

Development  of  Pascal's  and  Brianchon's  Theorems,        .         .         .  508 


BOOK   SECOND:  — CO-ORDINATES   IN   SPACE. 
CHAPTER    FIRST. 

THE  POINT. 

Rectangular  Co-ordinates  in  Space  explained,             ....  514 

Expressions  for   Point  on   either   Reference-plane; — for   Point 

on  either  Axis; — for  Origin,     ......  515 

Polar  Co-ordinates  in  Space,        ........  516 

The  doctrine  of  Projections,         .         .         .         .         .         .         .         .517 

Distance  between  any  Two  Points  in  Space,      .....  520 

Relation  between  the  Direction-cosines   of  any  Right  Line,       .  521 

Co-ordinates  of  Point  dividing  this  distance  in  Given  Ratio,   .         .  522 

Transformation  of  Co-ordinates  : 522 

I.  To  change  Origin,  Reference-planes  remaining  parallel  to 

first  position,        ........  522 

II.  To  change  Inclination  of  Reference-planes,         .         .         .  522 
III.  To   change    System  —  from   Planars   to   Polars,   and   con- 
versely,           523 

General  Principles  of  Interpretation  : 524 

I.  Single  equation  represents  a  Surface, 524 

II.   Two  equations  represent  Line  of  Section,     ....  524 

III.  Three  equations  represent  mnp  Points,          ....  524 

IV.  Eq.  wanting  abs.  term  represents  Surface  through  Origin,  .  524 
V.  Transf 'n  of  Co-ordinates  does  not  affect  Space-Locus,         .  524 


CHAPTER    SECOND. 

LOCUS  OF  FIRST  ORDER  IN  SPACE. 
Equation  of  First  Degree  in  Three  Variables  represents  a  Plane,     .     525 

General  Form  of  Equation  to  any  Plane, 526 

Equation  to  Plane  in  terms  of  its  Intercepts  on  Axes,        .         .         .     527 


CONTENTS.  xxxin 

PAGE. 

Equation  to   Plane  in   terms  of  its   Perpendicular  from  Origin  and 

Direction-cosines, 527 

Transformation  of  Ax  +  By  +  Cz  +  D  =  0  to  the  form  last  obtained,  528 

THE  PLANE  UNDER  SPECIAL  CONDITIONS,              529 

Equation  to  Plane  through  Three  Fixed  Points,          .         .         .  529 
Angle   between  two  Planes  :    conditions  that  they  be  par- 
allel or  perpendicular, 529 

Equation  to  Plane  parallel  to  given  Plane; — perpendicular  to 

given  Plane, 531 

Length  of  Perp'r  from  (xyz)  on  x  cos  a  -j-  y  cos  /?  +  z  cos  y  =  0,  532 

"                 "          "         "      on  Ax  +  By  +  Cz  +  D  =  0,     .         .  532 
Equation    to   Plane    through    Common    Section    of    two    given 

ones,  P+/cP'  =  0, 532 

Equation  to  Planar  Bisector  of  angle  between  any  two  Planes,  533 

Condition  that  Four  Points  lie  on  one  Plane,      ....  533 

"            "     Three  Planes  pass  through  one  Right  Line,       .  533 

"  "     Four  Planes  meet  in  one  Point,  .         .         .534 

QUADRIPLANAR  CO-ORDINATES:    Abridged  Notation  in  Space,    .         .  534 

la  +  m/3  +  ny  +  r6  =  0  represents  a  Plane  in  Quadriplanars,         .  535 

LINEAR  Loci  IN  SPACE:    solved  as  Common  Sections  of  Surfaces,     .  535 

The  Right  Line  in  Space  as  common  Section  of  Two  Planes,      .  535 

Equation  to  Right  Line  in  terms  of  Two  Projections,          .         .  536 

Symmetrical  Equations  to  the  Right  Line  in  Space,            .         .  537 

To  find  the  Direction-cosines  of  a  Right  Line,             .         .         .  537 

Angle  between  Two  Right  Lines  in  Space,           ....  538 

Conditions  as  to  Parallelism  and  Perpendicularity,     .         .         .  539 

Equation  to  Right  Line  perpendicular  to  given  Plane,        .         .  539 

Angle  between  a  Right  Line  and  a  Plane,           ....  540 

Condition  that  a  Right  Line  lie  wholly  in  given  Plane,       .         .  540 

Condition  that  Two  Right  Lines  in  Space  shall  intersect,   .         .  541 

Examples  involving  Equations  of  the  First  Degree,  ....  541 


CHAPTER    THIED. 

LOCUS  OF  SECOND  ORDER  IN  SPACE. 

General  Equation  of  Second  Degree  in  Three  Variables:  —  its  gen- 
eral interpretation,     .........     542 

SURFACES  OF  SECOND  ORDER  IN  GENERAL: 

Criterion  of  the  Form  of  any  Surface   furnished   by   its   Sections 

with  Plane, 543 

Every  Plane  Section  of  Second  Order  Surface  is  a  Curve  of  Sec- 
ond Order, 543 

Properties  common  to  all  Quadrics,  .....  544 — 552 
Classification  of  QUADRICS,  or  Surfaces  of  Second  Order,  .  .  553 
Summary  of  Analogies  between  Quadrics  and  Conies,  '  .  .  561 


xxxiv  CONTENTS. 

PAGE. 

SURFACES  OP  REVOLUTION  OF  THE  SECOND  ORDKR:      .        .        .        .  562 

General  Method  of  Revolutions  explained,           ....  563 

Equation  to  the  Cone,            ........  564 

Demonstration  that  all  Curves  of  Second  Order  are  Conies,         .  565 

Equation  to  the  Cylinder, 567 

"             "       Sphere, 567 

Equations  to  the  Ellipsoids  of  Revolution,           ....  568 

"             "         Hyperboloids          " 568 

The  Ellipse  of  the  Gorge,  and  its  Equation,       .         .  569 

"           to  the  Paraboloid, 569 

The  Tangent  and  Normal  Planes  to  the  Quadrics,     .         .         .         .570 

EXAMPLES, 573 


NOTE. 

The  references  in  the   present   treatise   are  to  RAT'S  New  Higher 
Algebra,  and  RAY'S    Geometry  and   Trigonometry. 


INTRODUCTION: 

THE  NATURE,  DIVISIONS,  AND  METHOD  OF 

ANALYTIC  GEOMETRY. 


ANALYTIC  GEOMETRY: 

ITS  NATURE,  DIVISIONS,  AND  METHOD. 


1.  BY  Analytic  Geometry  is  meant,  speaking  gen- 
erally, Geometry  treated  by  means  of  algebra.     That  is  to 
say,  in  this  branch  of  mathematics,  the   properties   of 
Figures,  instead  of  being  established  by  the  aid  of  dia- 
grams, are  investigated  by  means  of  the  symbols  and 
processes   of  algebra.     In    short,   analytic   is   taken    as 
equivalent  to  algebraic. 

2.  Accordingly,  and  within  recent  years  especially, 
the  science  has  sometimes  been  called  Algebraic  Geom- 
etry.    It  is  preferable,  however,  for  reasons  which  will 
appear  farther  on,  to  retain  the  older  and  more  usual 
name.      Why  algebraic  treatment  should  be  considered 
analytic,  —  in    what    precise    sense    geometry   is    called 
analytic  if  treated  by  algebra,  when  it  is  not  called  so 
if  treated  by  the  ordinary  method,  —  will  appear  as  wre 
proceed.     But,  for  the  present,  the  attention  needs  to  be 
fixed   upon   the    simple   fact,   that,  in   connection    with 
geometry,  analytic  means  algebraic. 

3.  The   properties   of  Figures   are   of  two  principal 
classes  :   they  either  refer-to  magnitude,  or  else  to  position 
and  form.     Thus,  The  areas  of  circles  are  to  each  other  as 
the  squares  upon  their  radii,  is  a  property  of  the  first 

(3) 


ANAL  YTIC  GEOMETR  Y. 

* 

kind ;    of  the  second  is,  Through  any  three  points,  not  in 
the  same  right  line,  one  circle,  and  but  one,  can  be  passed. 

4.  Accordingly,  geometric  problems  are  either  Prob- 
lems of  Dimension  or  Problems  of  Form. 

5.  Corresponding  to  these  two   classes  of  problems, 
there  are,  in  Analytic   Geometry,  two   main   divisions; 
namely,  DETERMINATE   GEOMETRY  and  INDETERMINATE 
GEOMETRY. 

The  methods  of  these  two  divisions  we  now  proceed  to 
sketch. 

I.   DETERMINATE  GEOMETRY. 

6.  The  geometry  of  Dimension  is  called  DETERMINATE, 
because  the  conditions  given  in  any  problem  in  which  a 
dimension  is  sought  must  be  sufficient  to  determine  the 
values  of  the  required  magnitudes ;  or,  to  speak  from  the 
algebraic  point  of  view,  these  conditions  are  always  such 
as  give  rise  to  a  group  of  independent  equations,  equal 
in  number  to  the  unknown  quantities  involved,  and  there- 
fore determinate.* 

7.  In  completing  a  problem  of  Determinate  Geometry, 
there  are  two  distinct  operations :   the  SOLUTION  and  the 
CONSTRUCTION. 

8.  The  Solution  for  the  required  parts,  consists  in 
representing  the  known  and  unknown  parts  of  the  figure 
in  question  by  proper  algebraic  symbols,  and  finding  the 
roots  of  the  equations  which  express  the  given  relations 
of  those  parts. 

The  Construction  of  the  parts  when  found,  consists 
in  drawing,  according  to  geometric  principles,  the  geo- 
metric equivalents  of  the  determined  roots. 


*  Algebra,  Art.  159,  compared  with  168. 


INTRODUCTION.  5 

The  principles  underlying  each  of  these  operations 
will  now  be  developed. 

PRINCIPLES    OF    NOTATION. 

9.  These  are  all  derived  from  the  algebraic  convention 
that  a  single  letter,  unaffected  with  exponent  or  index, 
shall  stand  for  a  single  dimension ;  or,  as  it  is  commonly 
put,  that  each  of  the  literal  factors  in  a  term  is  called  a 
dimension  of  the  term.  From  this,  it  follows  that  the 
degree  of  a  term  is  fixed  by  the  number  of  its  dimen- 
sions. Our  principles  therefore  are : 

1st.  Any  term  of  the  first  degree  denotes  a  LINE,  of  de- 
terminate length.  For,  by  the  convention  just  stated,  it 
denotes  a  quantity  of  one  dimension.  When  applied  to 
geometry,  therefore,  it  must  denote  a  magnitude  of  one 
dimension.  But  this  is  the  definition  of  a  line  of  fixed 
length.  Accordingly,  a,  x  denote  lines  whose  lengths 
have  the  same  ratio  to  their  unit  of  measure  that  a  and  x 
respectively  have  to  1. 

2d.  Any  term  of  the  second  degree  denotes  a  SURFACE, 
of  determinate  area.  For  it  denotes  a  magnitude  of  two 
dimensions,  that  is,  a  surface ;  and  since  each  of  its 
dimensions  denotes  a  line  of  fixed  length,  the  term,  as 
their  product,  must  denote  a  surface  of  equal  area  with 
the  rectangle  under  those  lines.  In  fact,  it  is  usually 
cited  as  their  rectangle.  Thus,  ab  denotes  the  rectangle 
under  the  lines  whose  lengths  are  a  and  b.  Similarly,  x2 
denotes  the  square  upon  a  side  whose  length  is  x. 

3d.  Any  term  of  the  third  degree  denotes  a  SOLID,  of 
determinate  volume.  For  it  is  the  product  of  the  lengths 
of  three  lines,  and  hence  denotes  a  volume  equal  to  that 
of  the  right  parallelepiped  between  those  lines,  and  is  so 
cited.  Thus,  abc  is  the  right  parallelepiped  whose  edges 


6  ANALYTIC  GEOMETRY. 

have  severally  the  lengths  a,  b,  c;  and  a?  denotes  the  cube 
on  the  edge  whose  length  is  x. 

4th.  An  abstract  number,  or  any  other  term  of  the  zero 
degree,  denotes  some  TRIGONOMETRIC  FUNCTION,  to  the 
radius  1.  The  general  symbol  for  a  term  of  the  zero 

a 

degree  may  be  written  7,  since  the  number  of  dimen- 
sions in  a  quotient  equals  the  number  in  the  dividend 
less  that  in  the    divisor.      If,  now, 
we  lay  off  any  right  line  AB  =  b, 
describe  a  semicircle  upon  it,  and, 
taking  the  chord  BC  =  a,  join  AC: 
we    shall   have    (Geom.,    Art.   225) 
the  triangle  ABC  right-angled  at  C, 
Hence,  (Trig.,  Art.  818,) 

-  =  sin  A. 
o 

If  the  base,  instead  of  the  hypotenuse,  were  taken  =  b, 
we  should  have 

a 

v-  =  tan  A. 

b 

If  the  hypotenuse  were  taken  =  a,  and  the  base  =  b, 
we  should  have 


-  =  sec  A ; 


and  so  on. 


5th.  A  polynomial,  in  geometric  use,  is  always  HOMO- 
GENEOUS, and  denotes  (according  to  its  degree)  a  length, 
an  area,  or  a  volume,  equal  to  THE  ALGEBRAIC  SUM  of 
the  magnitudes  denoted  by  its  terms.  By  the  ordinary 
convention  of  signs,  it  must  denote  the  sum  mentioned; 
it  is  therefore  necessarily  homogeneous,  since  the  sum- 


INTRODUCTION.  1 

mation  of  magnitudes  of  unlike  orders  is  impossible. 
We  can  not  add  a  length  to  an  area,  nor  an  area  to  a 
volume. 

Corollary. — Hence,  if  a  given  polynomial  be  apparently 
not  homogeneous,  it  is  because  one  or  more  of  the  linear 
dimensions  in  certain  of  its  terms  are  equal  to  the  unit 
of  measure,  and  consequently  represented  by  the  im- 
plicit factor  1.  WJien,  therefore,  such  a  polynomial  occurs, 
before  constructing  it,  render  its  homogeneity  apparent  by 
supplying  the  suppressed  factors.  Thus, 

a3  -f  a?b  —  c  —fg  =  a*  +  a2b  —  c  X  1  X  1  —fg  X  1 . 
Similarly,  for 


we  may  write 


and  so  on. 

6th.  Terms  of  higher  degrees  than  the  third  have  no 
geometric  equivalents.  For  no  magnitude  can  have  more 
than  three  dimensions. 

Corollary.  —  If  expressions  apparently  of  such  higher 
degrees  occur,  they  are  to  be  explained  by  assuming  1  as  a 
suppressed  divisor,  and  constructed  accordingly.  Thus, 


=' 

and  so  on. 


Remark — These  six  principles  enable  us  to  represent  by 
proper  symbols  the  several  parts  of  any  geometric  problem,  and 
to  interpret  the  result  of  its  solution,  as  indicating  a  line,  a 
rectangle,  a  parallelopiped,  etc.  We  then  construct  the  magni- 
tude thus  indicated,  according  to  the  principles  to  be  explained 
in  the  next  article. 
An.  Ge.  4. 


8  ANALYTIC  GEOMETRY. 

EXAMPLES. 

1.  Render  homogeneous  a?b  -\-  c —  d2. 

2.  In  what  different  Avays  may  the  degree  of  la  be  reckoned  ? 
Of  5xyl  Of  V 5 (#2-|-y2)?     State  their  geometric  meaning  for  each 
way. 

3.  Interpret  geometrically  i/2o6;   1/3;   V^z;  and  \^6a. 

4.  Adapt  a&  to  represent  a  line:   also,  i/ota. 


5  .  What  does  V  d1  +  62  represent  ?    What  V  m*  +  nz  —  I'2  —  r2  ? 


6.  vd  being  given  as  denoting  a  surface,  render  its  form  con- 
sistent with  its  meaning. 

7.  Render          c-  —  ~  homogeneous  of  the  second  degree. 

8.  Render  —  -  homogeneous  of  the  first  degree. 

9.  Render  -  —  —  —  ^  —  homogeneous  of  the  zero  degree. 

10.  Adapt  a5^2  to  represent  a  solid;  —  a  surface;  —  a  line.    What 
is  the  geometric  meaning  of  a46~2?   of  a56~x?  of  a~2? 


PRINCIPLES    OF    CONSTRUCTION. 

1O.  In  these,  we  shall  confine  ourselves  to  construct- 
ing the  roots  of  Simple  Equations  and  Quadratics. 

I.  The  Root  of  the  Simple  Equation. — This  may 
assume  certain  forms,  the  construction  of  which  can  be 
generalized.  The  following  are  the  most  important: 

1st.  Let  x  =  a  =L  b.  Here,  (Art.  9,  5)  x  denotes  a 
line  whose  length  is  the  algebraic  sum  of  those  denoted 
by  a  and  b.  Therefore,  on  any  right  line,  take  a  point  A 
as  the  starting-point,  or  Origin,  and  lay  off  (say  to  the 
right)  the  unit  of  measure  till 

AB  =  a.     Then,  if  b  is  pos-      —  -^ i — j — +r~ 

idee,    by    laying    off,    in    the 

same  direction  and  on  the  same  line  as  before,  JBC—b, 


INTRODUCTION. 

we  obtain  A  0  as  the  required  line ;  for  it  is  evidently 
the  sum  of  the  given  lengths.  But,  if  b  is  negative,  its 
effect  will  be  to  diminish  the  departure  from  A ;  hence, 
in  that  case,  it  must  be  laid  off  as  BD  =  BC  =  b.  We 
thus  obtain  AD  for  the  required  line  ;  and  it  is  obviously 
equal  to  the  difference  of  the  given  lengths. 

If  b  >  a  and  negative,  then  x  is  wholly  negative.  Our 
construction  answers  to  this  condition.  For  then  the 
extremity  of  b  will  fall  to  the  left  of  the  origin  A,  say  at 
E,  and  the  line  x  will  therefore  be  represented  by  AE, 
and  measured  wholly  to  the  left  of  A. 

This  brings  into  view  the  important  principle  that  the 
signs  -f-  and  —  are  the  symbols  of  measurement  in  opposite 
directions.  Hence,  if  we  have  a  linear  polynomial,  its 
negative  terms  are  to  be  constructed  by  retracing  such  a 
portion  of  the  distance  made  from  the  origin,  correspond- 
ing to  its  positive  terms,  as  their  length  requires.  If  we 
have  two  monomials  with  contrary  signs,  they  must  be 
laid  off  in  opposite  directions  from  the  origin. 

ab 

2d.    Let  x  =  —  .      In 
c 

this  case,  x  denotes  a  line 

whose  length  is  a  fourth 

proportional  to  c,  a,  and  b. 

Therefore,  draw  two  right 

lines,  AC  and  AE,  making  any  angle  with  each  other. 

On  the  one,  lay  off  AB  =c9  and  AC '  =  a;  on  the  other, 

AD  =  b.     Join  ED,  and  draw  CE  parallel  to  BD-  then 

will  AE  be  the   line  required.     For,    (Geom.,  307)  the 

triangles  ABD,  ACE  being  similar,  we  have 

AB  :  AC  ::  AD  :  AE;       or,     c  :  a  ::  b  :  AE. 


10  ANALYTIC  GEOMETRY. 

3d.    Let  x  =  -j-  .       Putting   -=  =  k,  this  may  be 

written 

ko 

x  =  —  . 
9 

ab 
Therefore,  construct  k  =  -^-,  as  in  the  preceding  case, 

and,  with  the  line  thus  found,  apply  the  same  construc- 

tion to  x. 

abed      k'd  abc      kc 

in  like  manner,    -  7  =  -7—  ,  by  putting;  k  =  —  =  —  • 
fgh        h  fy       9 

And,  in  the  same  way,  we  may  construct  any  quotient 
of  the  first  degree. 

II.  The  Roots  of  the  Pure  Quadratic.  —  Three  or 
four  cases  deserve  attention  : 

1st.  Let  x  =  V  ab.     Here  we  have  a  line  whose  length 
is  the  geometric  mean  of  a  and  b.     There  are  several 
constructions,  but   the   following  is  as   elegant  as  any. 
On  any  right  line,  lay  off  AB  =  a, 
and  BC  =  b.      Upon  AC  =  a  -f  6, 
describe  a  semicircle.     At  B  erect  a 
perpendicular  meeting  the  curve  in 
D:   BD    is  the   line    sought.     For 
(Geom.,  325) 

BD  =  I^ABXBC  =  Vdb  =  x. 


Striptly  speaking,  since  the  radical  V  ab  has  a  double 
sign,  there  are  two  lines  answering  to  x,  equal  in  length, 
but  measured  in  opposite  directions.  And  for  this,  in 
fact,  the  construction  provides:  since  there  is  a  semi- 
circle below,  as  well  as  above  AC,  to  which  the  per- 
pendicular dropped  from  B  has  the  same  length  as 
but  is  drawn  in  a  direction  exactly  opposite. 


ft.J.ef. 


1NTR  01)  UCTION. 


1  1 


2d.  Let  x  =  l/a2  -|-  ac.     Writing  this  in  the  form 


we  perceive  that  we  have  to   construct   the   geometric 

mean  of  a  and  a  -\-  c.     On  any  right 

line  lay  oiF  A  B  =  a,  and  B  C=  c  :  then 

A  0  —  -  a  -f  c.    Describe  a  semicircle 

about  AC,  erect  the  perpendicular 

BD,  and  join  AD:  AD  is  the  line 

required.     (Geom.,  324,  2.)     In  this 

case,  to  satisfy  the  negative  value  of  x,  we  must  lay  off 

a  and  c  from  A  to  the  left,  and   throw  our  semicircle 

below  the  —  (a  -f-  c)  thus  formed.     The  geometric  equiv- 

alent of  x  is  the  chord  joining  A  and  the  point  where 

the  perpendicular  from  the  extremity  of  —  a  meets  this 

downward  semicircle.     And  this  chord  is  obviously  drawn 

from  A  in  exactly  the  opposite  direction  to  AD. 

3d.  Let  x  =  Va'2jrb2.  This  gives  us 
the  side  of  a  square  whose  area  equals 
the  sum  of  two  given  squares.  Ac- 
cordingly, lay  off  AB  =  a  ;  at  B  erect 
a  perpendicular,  and  upon  it  take 
BC  =  b;  join  A  C:  then  (Geom.,  408) 
AC  is  the  line  required. 

4th.  Let  x  =  V/~^r—  b2.     In  this  case,  we  are  to  con- 
struct the  side  of  a  square  whose  area  equals  the  differ- 
ence of  two  given  squares.    Lay  off 
AB  =  c,  and  erect  upon  it  a  semi- 
circle.    From  A  as  a  center,  with  a 
radius  A  C  =  b,  describe  an  arc  cut- 
ting the  semicircle  in  C.     Join   CB, 
arid  (Geom.,  409)  BO  will  be  the  required  line. 

We  leave  the  discussion  of  the  negative  values  of  x,  in 
the  last  two  cases,  to  the  student. 


12  ANALYTIC  GEOMETRY. 

III.   Roots    of    the    Complete    Quadratic.  —  Let 

us    consider    the   Four   Forms    separately.     (See   Alg., 
231.) 

1st.  The  First  Form  of  the  complete  Quadratic  is 


Its  roots  are  given  in  the  formula 


x  =  —  p  ±     f  -f  f. 

To  construct  these :  Lay  off  AB  =  q.  At  B  erect  the 
perpendicular  BC  =  p.  From 
G  as  center,  with  the  radius  BQ 
describe  a  circle.  Join  AC,  and 
produce  the  line  to  meet  the  cir- 
cle in  E:  AD  and  EA  are  the 
required  roots.  For,  by  the  con- 
construction,  AC  =  V p2  -f-  <f '"•) 

and,  the  first  of  the  above  roots  having  the  radical  positive, 
we  must  lay  it  off  to  the  right,  from  A  to  C.  The  neg- 
ative p  must  now  be  laid  off  to  the  left,  from  C  to  D. 
Hence, 

AD  =    -p  +  Vp'  +  <f  =  d  (1). 

In  the  second  of  the  above  roots,  the  radical  is  negative. 
Both  that  and  p  must  therefore  be  measured  to  the  left 
from  their  origin.     We  begin,  then,  at  E  and  lay  off 
EC  =    -p;    continuing  in   the   same   direction,   CA  = 
-  Vf  -\-  q2.     Whence 


EA  =    -p  —  Vp2jrq'2  =  x"  (2). 

2d.  The  Second  Form  is  written 
x-  —  2px  ==  if. 


*"By  writing  the  absolute  term  as  of  the  2d  degree,  since  (Art.  9,  5)  the 
equation  is  homogeneous. 


INTE  OD  UCTION.  13 

Solving  for  x7  we  obtain 

x=p  ±:  V p2  -\-  (f. 

The  construction  is  the  same  as  in  the  First  Form,  ex- 
cept that  the  roots  are  laid  off  differently.  In  the  first 
root,  p  and  the  radical  are  both  positive;  accordingly, 
we  begin  at  A,  and  take  i/p2  +  <f  =  A  C;  p  must  be 
taken  in  the  same  direction  =  CE:  hence  we  obtain 


In  the  second  root,  p  being  positive  and  the  radical  neg- 
ative, we  begin  at  D,  and  take  p  =  DC;  we  then  retrace 
our  steps,  taking  —  Vp2  -f  q2  =  CA.  Whence 

DA=p  —  Vf  -\-q'2  =  x"  (4). 

3d.  The  Third  Form  is 


whence 

x=  -p±.Vf  —  <f. 

We  construct  these  roots  as  follows :    On  any  right  line, 

lay  off  AB  =  p ;  erect  at  B  a  perpendicular,  and  take 

upon  it  BC  =  q;  from  C  as  a  center, 

with  a  radius  equal  to  p,  describe  an  j 

arc  intersecting  AF  in  D  and  E: 

DA  and  EA  are  the  roots  required.       x  /' 

For,   by    the    construction,    DB  = 

BE=  V/p2  —  q2.    From  D  as  origin, 

take   DB  -  -  Vp2  -  —  q2 ;    from   B,   measure    backwards 

B  A  =  —  p :  and  we  obtain 


DA=    -p  +  Vf^cf  =  x'  (5). 

If  from   E  as   origin   we   measure  to   the  left,  EB  = 
—  ~\/p2—q2,  and  B A  =  — p.     Whence  we  have 

EA  =  —  p  —  Vf  —  cf  =  x"  (6). 


14  ANALYTIC  GEOMETRY. 

4th.  The  Fourth  Form  is 


x2  —  Qpx  =  —  q2. 


Its  roots  are 


x=p±  V  p2 — q- 


Using  the  same  general  construction  as  in  the  Third 
Form,  we  find  the  linear  equivalent  of  the  first  of  these 
by  assuming  A  as  origin,  taking  AB  =  p  to  the  right, 
and,  in  the  same  direction,  BE  =  Vp2  —  q2.  Whence, 

-f^  (7). 


For  the  second  root,  still  making  A  the  origin,  we  have 
AB  =  p,  and  BD  =  —  i/p2  —  q2.     Therefore, 


AD  =  p  —  i/p2  —  q2  =  x"  (8). 

Remarks. — The  constructions  just  explained  furnish 
a  good  example  of  the  clearness  and  completeness  with 
which  algebraic  and  geometric  properties  reflect  each 
other,  when  the  necessary  conventions  are  established. 
Thus, 

First:  The  construction  in  the  First  Form  reflects  the 
algebraic  property  (Alg.  234,  Prop.  4th)  that  the  absolute 
term  of  a  complete  quadratic  is  equal  to  the  product  of 
its  roots.*  For,  by  the  construction,  AB  =  q  is  tangent 
to  the  circle  DBE  at  B.  Hence  (Geom.,  333)  AB2  = 
AD  X  AE.  That  is,  cf  =  x'x". 

On  the  other  hand,  we  may  see  that  the  algebraic  con- 
dition, q2  =  x'x" ,  gives  us  the  geometric  property  that 
the  square  on  the  tangent  to  a  circle,  from  any  point  with- 
out the  curve,  is  equal  to  the  rectangle  under  the  segments 


*  We  assume  here,  as  we  shall  generally  throughout  the  book,  that 
the  absolute  term  is  written  in  the  first  member  of  the  equation. 


INTR  OD  UCTION.  1  5 

of  the  corresponding  secant.      For,  multiplying  together 
equations  (1)  and  (2),  we  have, 


But  our  condition  gives  us  xfx"  —  q2;   and,  by  the  con- 
struction, cf-  =  AB2.     Hence, 


Now,  AB  and  AE  are  respectively  the  tangent  and 
secant  from  A  to  the  circle  DBF. 

Second:  If  we  compare  equations  (1)  and  (4),  (2)  and 
(3),  we  observe  that  the  linear  equivalents  of  xf  in  (1) 
and  x"  in  (4),  of  x"  in  (2)  and  x'  in  (3),  are  identical 
lengths,  measured  in  opposite  directions.  In  other  words, 
the  positive  root  of  the  First  Form  is  the  negative  root 
of  the  Second,  and  vice  versa.  This  is  as  it  should  be: 
for,  obviously,  the  First  Form  becomes  the  Second,  if 
we  put  —  x  for  -f-  x. 

Third:  If,  in  equations  (5),  (6),  (7),  (8),  we  suppose 
p  >>  q-,  the  roots  are  real  and  unequal.  The  construcr 
tion  also  indicates  this.  For,  so  long  as  the  hypotenuse 
CD=p  is  greater  than  the  perpendicular  CB  =  q,  it  will 
intersect  AF  in  two  real  points. 

If  we  suppose  p  =  q,  the  roots  are  real,  but  equal. 
This,  too,  is  involved  in  the  construction.  For,  when  the 
radius  CD=p  becomes  equal  to  CB  =  q,  the  circle  touches 
AF;  that  is,  its  two  points  of  section  with  AF  become 
coincident  in  B,  and  AD  =  AB  =  AE. 

Fourth:  If,  in  (5),  (6),  (7),  (8),  we  suppose  q  >  p,  the 
roots  become  imaginary.  With  this,  again,  the  construc- 
tion perfectly  agrees.  For,  if  q  >  p,  the  radius  CD  is 
less  than  the  perpendicular  CB,  and  the  circle  cuts  AF 
in  imaginary  points.  The  supposition,  moreover,  re- 
quires us  to  construct  a  right  triangle  whose  hypotenuse 
An.  Ge.  5. 


16  ANALYTIC   GEOMETRY. 

shall  be  less  than  its  perpendicular  —  a  geometric  impos- 
sibility. This  agrees  with  the  well-known  algebraic 
principle,  that  imaginary  roots  arise  out  of  some  incon- 
gruity in  the  conditions  upon  which  the  equation  is 
founded. 

EXAMPLES. 

1.  Construct  x  =  V  I'1  +  m2  —  n2,  first,  by  placing  m2  —  n2  =  k'2; 
secondly,  by  placing  I'* -\- m*  —  Tc1 ;  thirdly,  by  placing  I'2  —  n2  =  &2. 
Show  that  the  three  constructions  give  the  same  line. 

Imn  -4-  &2A 

2.  Construct  x  =  —    . 

3.  Construct  v'5,  l/3afo,  and  >/a2-j-62- 

Imn  —  &2A  * 

4.  Construct  x  =  —       . 


5.  Construct  o?  = -\/ it-*/—  (£2 —  mn). 

\   n  \  n^ 


DETERMINATE    PROBLEMS. 

11.  The  mode  of  applying  the  foregoing  principles  to 
the  solution  of  these  problems,  may  be  best  exhibited  in 
a  few  examples. 

EXAMPLES. 

1.   In  a  given  triangle,  to  inscribe  a  square. — A  triangle  is  given 
when  its  base  and  altitude  are  given;    we  are  therefore  here  re- 
quired to  find  the  side  of  the  inscribed  square 
in  terms  of  the  base  and  altitude  of  the  given 
triangle.     If  we  draw  the  annexed  diagram, 
representing  the  problem  as  if  solved,  and 
designate  the  base  of  the  triangle  by  b,  its 
altitude  by  A,  and  the  side  of  the  inscribed 
square  by  x:  then,  since  the  triangles  CAB, 
CEF  are  similar,  we  have  (Geom.,  310) 


*  The  student  should  pay  strict  attention  to  the  geometric  meaning 
of  the  signs  +  and  — ,  as  explained  above.  He  should  also  see  that  the 
given  algebraic  expressions  are  put  into  the  most  convenient  forms,  before 
constructing. 


INTRODUCTION. 


17 


AB  :  EF  : :  CD  :  Off;     or,    b  :  x  :  •  h  :  A—  x. 

bh 
.' .    x  =  . 

That  is,  the  side  of  the  inscribed  square  is  a  fourth  proportional  to 
the  base,  the  altitude,  and  their  sum.  We  therefore  construct  it  as 
in  Art.  10,  I.  2d.  Or  it  may  be  more  conveniently  done  as  follows: 
Produce  the  base  of  the  given  tri- 
angle until  BL  —  A ;  through  L 
draw  LM,  parallel  and  equal  to 
BC;  join  MAt  and  from  JV,  the 
point  where  MA  cuts  BC,  drop  a 
perpendicular  upon  AB:  then  is 
NO  the  side  required.  For,  letting 
fall  MP  perpendicular  to  AL,  we  have,  by  the  similar  triangles 
MAL,  NAB, 

AL  :  AB  ::  MP  :  NO;     that  is,    b  +  h  :  b  : :  A  :  NO. 

bh 


\ 


Note. — The  student  should  consider  what  several  positions  the 
side  of  the  square  may  assume  according  as  the  triangle  is  acute- 
angled,  right-angled,  or  obtuse-angled. 

2.  In  a  given  triangle,  to  inscribe  a  rectangle  whose  sides  are  in  a 
given  ratio. — Let  x  and  y  represent  the  two  sides,  and  r  their  con- 
stant ratio.  Then  we  shall  have 


x 


(1). 


And,  as  in  the  previous  example,  (the  other 
symbols  remaining  the  same,)  we  obtain  the 
proportion  b  :  y  :  :  A  :  A  —  x. 

.-.    hy  =  bh  —  bx  (2). 

Eliminating  y  between  (1)  and  (2), 

bh 


This  value  we  construct  in  the  same  manner  as  the  side  of  the 
inscribed  square.  In  fact,  as  is  obvious,  the  first  problem  is  merely 
a  particular  case  of  the  present  one  ;  for  a  square  is  a  rectangle,  the 
ratio  between  the  sides  of  which  is  equal  to  1.  The  solution,  too, 


18 


ANALYTIC  GEOMETRY. 


shows  this ;  for  the  value  of  x  in  the  present  example  becomes  that 
obtained  in  the  former,  when  r  —  1. 

Produce,  then,  the  base  of  the  given  triangle  until  BL  equals 
rh ;  and  complete  the 

drawing    exactly  as    in  C  _M 

the  case  of  the  inscribed 
square  :  the  point  -ZV,  in 
which  the  diagonal  AM 
cuts  the  side  of  the  tri- 
angle, is  a  vertex  of  the 
required  rectangle.  Let  the  student  prove  this. 

3.  To  draw  a  common  tangent  to  two  given  circles. — Here  our  data 
are  the  radii  of  the  circles,  and  the  distance  between  their  centers. 
Let  r  denote  the  radius  of  the  circle  on  the  left,  and  r'  that  of  the 
other.  Let  d  =  the  distance  between  the  centers. 

The  problem  may  be  otherwise  stated :  Required  a  point,  from 
which,  if  a  tangent  be  drawn  to  one  of  two  given  circles,  it  will  also 
touch  the  other.  From  the  method  of  constructing  a  tangent,  (see 
©eom.,  230,)  it  follows 
that  this  point  is  some- 
where on  the  line  join- 
ing the  centers.  Hence, 
drawing  the  diagram  as 
annexed.,  it  is  evident 
that  our  unknown  quan- 
tity is  the  intercept  made 
by  the  tangent  on  this  line : 


that  is,  we  let 
x=  CT. 


Now  we  have  (Geom.,  333) 


In  like  manner, 

M'T*  =  N'TXI/T;  or,  M'T*=  (x  —  d  +  r^)  (x  —  d  -r')   (2). 
Expanding,  and  dividing  (1)  by  (2), 

MT*  xt  —  r* 


(3). 


But,  by  similar  triangles, 
MT  : 


::   r  :   7'     .-. 


MT< 


INTE  OD  UCTION.  19 


Substituting  in  (3),  and  reducing, 

(r2  —  r'2)  x2  —  2r2dx  +  r*dz  =  0  : 


Before  constructing  this  result,  let  us  interpret  it.  We  observe 
that  our  problem  involves  the  solution  of  a  quadratic,  and  that  we 
thus  obtain  a  double  value  for  x  =  CT.  The  required  tangent, 
therefore,  cuts  the  line  of  the  centers  in  two  points;  that  is,  there 
are  two  points  from  which  a  common  tangent  to  the  two  circles 
may  be  drawn.* 

Let  us  now  consider  the  two  values  of  x  more  minutely.  We 
shall  find  all  the  geometric  facts  of  the  problem  perfectly  repre- 
sented in  them. 

First  take  the  value  numerically  the  greater,  namely, 

rd 

X=7=?'' 

Since  this  must  be  numerically  greater  than  rf,  and  since  the 
definition  of  a  tangent  renders  it  impossible  that  the  point  sought 
should  fall  within  either  of  the  circles,  the  point  determined  by  this 
value  of  x  is  beyond  both.  If  r  ^>  r*,  x  is  positive,  and  the  point  T 
falls  to  the  right  of  both  circles,  as  in  the  diagram;  if  r</y,  a;  is 
negative,  and  the  point  then  falls  to  the  left  of  both. 

Secondly,  the  value  numerically  the  less, 


This  is  numerically  less  than  rf,  and,  in  connection  with  the  defi- 
nition of  the  tangent,  indicates  that  the  corresponding  point  lies 
between  the  two  circles  —  a  fact  with  which  the  sign  of  x  agrees: 
for  the  present  value  being  necessarily  positive  places  the  point  to 
the  right  of  C. 

We  learn,  then,  from  this  analysis,  that  (1)  there  are  two  points 
which  satisfy  the  conditions  of  the  problem;  that  (2)  one  lies 
beyond  both  circles,  and  the  other  between  the  two;  that  (3)  the 
former  falls  to  the  right  of  both  circles,  or  to  the  left  of  both,  ac- 
cording as  the  circle  whose  center  is  taken  as  the  origin  has  a 


*  Of  course,  there  are  four  common  tangents  —  two  tangents  from 
any  given  point  to  a  circle  being  always  possible.  But  as  these  exist  in 
pairs,  all  the  analytic  conditions  will  be  exhausted  in  two.  Hence,  we 
have  a  quadratic  to  solve,  rather  than  an  equation  of  the  fourth  degree. 


20  ANALYTIC  GEOMETRY. 

greater  or  less  radius  than  the  other.  How  perfectly  all  this  agrees 
with  the  geometric  conditions,  is  manifest.  By  merely  inspecting 
the  diagram,  we  can  see  that  two  common  tangents  can  be  drawn, 
one  passing  without  both  circles,  and  intersecting  the  line  of  the 
centers  beyond  the  smaller  circle;  the  other  passing  between 
the  two. 

Resuming  now  the  general  expression, 

rd 

-7±~7: 

if  we  suppose  r  =  /  we  obtain  x  =  oo  and  x  =  T) ;  from  which  we 
learn  that,  in  the  case  of  two  equal  circles,  the  external  tangent  is 
parallel  to  the  line  of  the  centers,  while  the  internal  tangent  bisects 
the  distance  between  the  centers.  This,  again,  obviously  accords 
with  the  geometric  conditions. 

If  r  —  0,  x  vanishes  for  both  its  forms:  hence,  in  this  case, 
the  two  tangents  are  drawn  from  the  center  which  was  assumed  as 
origin,  and  are  coincident.  This  should  be  so,  since,  if  r  =  0,  the 
corresponding  circle  is  reduced  to  a  point,  and  we  have  the  ordinary 
problem  of  the  tangent  to  a  circle  from  a  given  point  without. 

If  r'  =  0,  the.  two  values  of  x  again  coincide,  and  x  =  d.  In 
this  case,  therefore,  the  problem  is  reduced,  as  before,  to  that  of  the 
tangent  from  a  given  point ;  but  the  point  is  now  the  vanished  sec- 
ond circle. 

If  r  =  0  =  ?y,  we  have  #  =  — ;   that  is,  the  required  tangent 

may  be  drawn  from  any  point  in  the  line  of  the  centers.  This,  too, 
is  as  it  should  be;  for,  when  both  circles  are  reduced  to  points, 
the  two  tangents  coincide  with  each  other  and  with  the  line  of  the 
centers;  and  a  line  coincident  with  a  given  line  may  always  be 
drawn  from  any  point  in  the  latter. 

Passing  now  to  the    construction  of  the  intercept  represented 

by  x  —       .    s ,  we  see  that  we  are  to  find  a  fourth  proportional  to 

r  =fc  /,  r,  and  d.  We  shall 
obtain  this  most  simply  as 
follows :  Draw  any  set  of 
parallel  radii,  as  CK.  C?Kf, 
producing  the  latter  to  meet 
its  circle  in  K".  Through 
(7f,  K'}  and  (K,  K")  draw 


INTR  OD  UCTION.  21 

right  lines:  the  points  T  and  T',  in  which  these  intersect  the  line 
of  the  centers,  are  the  extremities  of  the  required  intercepts.  For, 
drawing  K'H  and  K"G  parallel  to  CT,  we  have,  by  similar  tri- 
angles, 


Whence, 

rd 


r-r"  r  +  S 

Therefore,  draw  through  T,  or  T',  a  tangent  to  either  of  the 
given  circles,  and  the  construction  is  complete.  * 

4.  To  construct  a  rectangle,  given  its  area  and  the  difference  between 
its  sides. — This  problem  is  of  importance,  as  illustrating  the  fact 
that  we  are  not  always  to  interpret  the  presence  of  a  quadratic  in 
our  investigations  as  indicating  a  double  solution  of  the  problem 
in  hand.  On  the  contrary,  a  quadratic  not  unfrequently  arises 
when  but  one  solution  is  possible.  One  of  its  most  important 
interpretations  in  that  case,  will  appear  in  solving  the  present 
example. 

Let  (Art.  9,  2)  a2  =  the  given  area  of  the  rectangle,  and  d  = 
the  difference  between  its  sides.  Let  x  =  the  less  side;  then  will 
x  -\-  d  =  the  greater. 

By  the  data  (Geom.,  379)  we  have 


a;     or,   xx  =  a. 
.'.    x  =  —  -  ±  A/ a2 -j- -j-  =  the  less  side. 


*  This  construction,  so  well  adapted  for  analytic  discussion,  some- 
times fails  in  practice,  as  the  extremity  of  the  outer  intercept  may  not 
fall  upon  the  paper.  The  following  elegant  construction  is  practicable  in 
all  cases  : 

From  the  center  of  the  larger  circle,  with  a  radius  equal  to  the 
difference  between  the  given  radii,  describe  a  circle,  to  which  draw  a 
tangent  from  the  center  of  the  smaller  circle.  A  tangent  to  either  given 
circle,  parallel  to  this,  is  tangent  to  both. 


22 


ANALYTIC  GEOMETRY. 


This  is  a  case  of  the  roots  of  a 
quadratic  in  the  First  Form.  We 
therefore  construct  x  as  in  Art.  10, 
III,  1.  We  then  have,  taking  the 
vipper  sign  in  the  values  of  both 
sides, 


d ,    r,    d-' 

=  -^^a  +T=-*' 


—  =  a:,  the  less  side ; 

AE  =  ;y  +  -yla'2  +  —  =  a;  +  rf,  the  greater. 

K  we  take  the  lower  sign  in  the  values  of  the  two  sides,  we  obtain 

d 


Comparing  with  the  former  values,  we  see  that  the  present  less  side 
is  the  negative  of  the  former  greater  ;  and  the  present  greater,  the 
negative  of  the  former  less.  Hence,  in  this  case, 

—  AE  —  the  less  side. 

—  AD  =  the  greater. 

It  is  obvious  that  the  expressions  less  and  greater  are  here 
used  in  their  algebraic  sense;  for  AE  is  still  numerically  greater 
than  AD. 

Now,  by  the  construction,  (Geom.,  333)  the  rectangle  of  the 
parts  gives  us 


Hence,  in  both  cases,  the  rectangle  is  positive,  and  absolutely  the 
same.  The  quadratic,  therefore,  does  not  here  indicate  two  solu- 
tions. It  merely  signifies  that  the  required  rectangle  may  be 
obtained  either  by  representing  its  sides  by  x  and  x  -f-  o?,  or  by 
—  x  and  —  (x  +  d).  That  is.  it  points  not  to  tAvo  rectangles  an- 
swering the  given  conditions,  but  merely  to  two  correlated  modes 
of  expressing  the  conditions  of  one  and  the  same  rectangle. 

We  learn,  then,  that  the  algebraic  discussion  of  a  problem  not 
only  possesses  the  greatest  generality  —  indicating  by  the  equa- 
tions to  which  the  problem  gives  rise  every  possible  solution;  but 


INTE  OD  UCTION.  23 

that,  if  there  are  various  modes  of  expressing  conditions,  which 
still  lead  to  the  same  equation,  the  equation  formed  on  the  basis 
of  any  one  of  these  modes  will  include  all  of  them  in  the  form  of 
its  roots. 

12.  From  the  foregoing  examples,  we  gather  the  fol- 
lowing rule  for  solving  Determinate  Problems: 

Draw  a  diagram  representing  the  problem  as  if  solved, 
inserting  any  auxiliary  lines  needed  to  develop  the  rela- 
tions between  the  known  and  unknown  parts.  By  means 
of  the  geometric  properties  which  the  diagram  involves, 
form  equations  between  these  parts,  taking  care  that  they 
be  independent  and  equal  in  number  to  the  unknown 
quantities.  Construct  upon  a  single  figure  the  roots  of 
these  equations. 

A  few  exercises  are  added,  which  the  beginner  should 
carefully  perform.  In  each,  let  the  problem  be  dis- 
cussed, as  to  the  number  of  its  solutions,  their  various 
meanings,  etc.  In  the  construction,  select  that  method 
which  is  neatest  and  most  convenient. 

EXAMPLES. 

1.  To  construct  a  square  of  equal  area  with  a  given  rectangle. 

2.  In  a  given  triangle,  to  inscribe  a  rectangle  of  a  given  area. 

3.  In  a  given  semicircle,  to  inscribe  a  square. 

4.  To  draw,  parallel  to  the  base  of  a  triangle,  a  line  which  shall 
divide  it  into  two  parts  equal  in  area. 

5.  Through   a  given  point  without  a  circle  to  draw  a   secant 
whose  internal  segment  shall  be  equal  to  a  given  line. 

C.   To  describe  a  circle  equal  in  area  to  two  given  ones. 

7.  To  draw,  from  a  given  line  to  a  given  circle,  a  tangent  of  a 
given  length. 

8.  To  draw,  from  a  given  line  to  a  given  circle,  the  tangent  of  the 
least  length. 


24  ANALYTIC   GEOMETRY. 

9.   Through  two  given  points,  to  describe  a  circle  touching  a 
given  right  line. 

10.  In  a  given  circle,  to  inscribe  three  equal  circles  touching 
each  other  externally. 

II.   INDETERMINATE  GEOMETRY. 

13.  The  geometry  of  Form  is  called  INDETERMINATE, 
because  all  Forms  are  conceived  to  arise  out  of  the  rela- 
tive positions  of  points ;  that  is,  out  of  a  point's  being 
so  far  indeterminate  as  to  be  capable  of  assuming  any  one 
of  a  series  of  positions  which  define  a  Form  :   or,  from 
the  algebraic  point  of  view,  because  the  equations  which 
express  the  conditions  under  which  a  point  may  vary  its 
position,  are  always  found  to  be  less  in  number  than  the 
unknown  quantities  they  contain,  and  hence,  admitting 
of  an  infinite  number  of  values  for  these,  are  indeter- 
minate. * 

14.  It  is  in  this  second  main  division  of  the  subject, 
that  we   come    upon    the  proper  province    of  Analytic 
Geometry.     In  fact,  as  the  student  has  doubtless  already 
noticed,  the  method  of  Determinate  Geometry  is  rather 
that  of  ordinary  geometry  than  of  algebra:   the  reason- 
ing is  based  mainly  on  the  diagram,  and  the  only  use 
of  the  algebraic   symbol  is  to  abbreviate  the  terms  of 
ordinary  language.     But,  in  the  geometry  of  Form,  as 
we  shall  soon  discover,  the  method  is  really  analytic: 
the   reasoning   is    strictly   algebraic,  while   the   symbol 
has   assumed  a   meaning  and   power  entirely  new.     In 
the    articles    immediately  following,    we    will    endeavor, 
first,  to  show  and  establish  the  fundamental  principle  of 
this  Geometry  of  Form,  or  of  Analytic  Geometry  strictly 
so  called;   secondly,  to  explain  in  outline  its  method  and 


*  Aig.,  168. 


INTRODUCTION.  25 

the  reasons  for  calling  it  analytic;  and  thirdly,  to  un- 
fold its  several  subdivisions,  especially  those  discussed 
in  the  body  of  the  present  work. 

I.    DEVELOPMENT    OF    THE    FUNDAMENTAL    PRINCIPLE. 

15.  The  principle  upon  which  the  whole  method  of 
Indeterminate  Geometry  is  founded  is  this:    The  alge- 
braic symbol  of  geometric  form  is  the  Equation. 

16.  The  figure  of  any  magnitude  is  obviously  deter- 
mined by  that  of  its  boundaries.     Hence,  all  Forms  are 
either  surfaces  or  lines.     If,  then,  we  can  show  that  an 
equation,    geometrically    interpreted,    represents    either 
some  line,  or  else  some  surface,  the  fundamental  prin- 
ciple will  be  established. 

*  17.  There  is,  of  course,  no  necessary  connection  be- 
tween the  symbols  of  algebra  and  the  conceptions  of 
geometry :  the  former  are  merely  conventional  marks, 
denoting  magnitudes  and  operations;  while  the  latter  are 
forms,  which  can  be  imagined  and  pictured,  and  which 
are  necessarily  the  same  to  every  mind. 

18.  The  truth  of  the  proposition  in  Art.  15  is  ac- 
cordingly not  necessary,  but  must  depend  upon  certain 
arbitrary  assumptions.  In  other  words,  if  the  symbols 
of  algebra  are  to  be  applied  to  represent  lines  and 
surfaces  —  if  symbols  of  magnitude  are  to  be  converted 
into  symbols  of  form  —  we  must  introduce  some  conven- 
tion as  to  their  meaning  in  the  new  connection.  This 
convention,  summarily  stated,  consists  in  making  the 
algebraic  symbols  of  magnitude  denote  the  distances,  linear 
or  angular,  of  a  point  from  certain  assumed  limits.  One 
form  of  it,  the  most  important  and  characteristic,  we  will 
now  illustrate,  confining  ourselves,  for  the  sake  of  sim- 
plicity, to  points  in  a  given  plane. 


26  ANALYTIC  GEOMETRY. 

19.  The  Convention  of  Co-ordinates. — Let  X'X, 

YY,  be  any  two  intersecting  right  lines,  having  any 
extent  we  please.  From  P,  any 
point  in  their  plane,  draw  PN 
parallel  to  the  first,  and  PM  par- 
allel to  the  second.  If,  now,  we 
assume  X'X  and  I^I^as  the  fixed 
limits  to  which  all  positions  in 
their  plane  shall  be  referred,  it  is 
obvious  that  we  know  the  position 
of  P,  so  soon  as  we  know  the  distances  NP  (or  its  equal 
OM)  and  MP.  Hence,  if  we  know  the  distances  of  any 
point  from  the  two  fixed  limits,  and  the  directions  in 
which  they  are  measured  from  these,  we  know  the 
position  of  the  point.  Accordingly,  if  we  can  repre- 
sent those  distances  and  directions  algebraically,  we  can 
represent  the  point. 

This  simple  apparatus  therefore  enables  us  at  once  to 
convert  the  algebraic  symbols  of  magnitude  and  direction 
into  symbols  of  position.  For  we  have  only  to  represent 
the  lengths  corresponding  to  OM  and  MP  by  letters, 
and  their  directions,  upward  or  downward  from  XX, 
to  the  right  or  to  the  left  from  Y'Y,  by  the  signs  -j- 
and  — .  The  letters  are  applied  according  to  the  con- 
ventions for  notation  given  in  Art.  9 ;  and  the  signs, 
according  to  the  usage,  familiar  in  trigonometry,  that  -j- 
shall  denote  measurement  upward  from  X'X  or  to  the 
right  from  Y'  Y,  and  -  -  measurement  downward  from 
X'X  or  to  the  left  from  YY.  Thus, 

-\-  a  =  OM,  with  -f  b  =  MP,  represents  P; 
-a  =  OM',  with  +  1>=M'P,  represents  P; 
-a  =  OM',  with  —l=M'P",  represents  P" ; 
+  a  =  OM,   with  —  b  =  MP'",  represents  P'". 


INTRODUCTION.  27 

The  lines  X'X,  Y'Y  are  called  axes;  their  intersec- 
tion 0  is  called  the  origin;  OM,  MP,  etc.,  are  called 
co-ordinates. 

If,  now,  instead  of  the  particular  lines  OM,  MP,  etc., 
we  take  x  and  y  as  general  symbols  for  the  co-ordinates 
of  a  point,  and  denote  by  a  and  b  the  values  they  as- 
sume for  any  particular  point,  we  obtain  the  algebraic 
expression  for  a  determinate  point  in  a  given  plane, 
namely, 

x  =  a 


in  which  a  and  b  may  have  any  value  from  0  to  oo,  and 
be  either  positive  or  negative. 

As  already  hinted,  the  foregoing  is  only  one  form  of  the 
convention  upon  which  rests  the  whole  structure  of  the 
geometry  of  Form.  Several  others  are  used,  differing 
from  the  present  in  the  nature  of  the  assumed  limits 
and  of  the  means  by  which  the  point  is  referred  to 
them  :  in  some,  as  in  the  present  one,  the  co-ordinates 
are  linear;  in  others,  one  of  them  is  angular.  The 
present  form,  moreover,  applies  only  to  points  in  a  given 
plane;  forms  suitable  for  representing  a  point  any- 
where in  space,  are  obtained  by  assuming  for  Fixed 
Limits  planes  instead  of  lines.  But  whether  the  forms 
of  the  convention  apply  to  points  in  space  or  to  points 
in  a  given  plane,  and  however  they  may  differ  in  their 
details,  they  all  agree  in  this:  that  a  point  shall  be  de- 
termined by  referring  it  to  certain  Fixed  Limits  by 
means  of  certain  Elements  of  Reference,  called  Co- 
ordinates. 

Definition.  —  The  Co-ordinates  of  a  point  are  it§ 
linear  or  angular  distances  from  certain  assumed  limits. 


28  ANALYTIC  GEOMETRY. 

Corollary. — By  the  Convention  of  Co-ordinates 

we  therefore  mean,  The  agreement  that  the  algebraic 
symbols  of  magnitude  shall  denote  the  co-ordinates  of  a 
point. 

20.  This  convention  once  established,  the  connection 
between  an  equation  and  a  geometric  form  will  readily 
become  apparent.     The  discovery  of  this  connection,  in 
its  universal  bearing,  and  the  first  exhaustive  applica- 
tion of  it  to  the  discussion  of  curves,  was  the  work  of 
the  French  philosopher  DESCAETES.     His  method  was 
first  published  in  1637,  in  his  treatise  De  la  Geometric. 
We  shall  now  show  that  the  connection  alluded  to  really 
exists;    but   must    first    define    certain    conceptions    on 
which  it  depends. 

21.  Variables  and  Constants. — In  analytic  inves- 
tigations, the  quantities  considered  are  of  two  classes: 
variables  and  constants. 

Definition. — A  Variable  is  a  quantity  susceptible,  in 
a  given  connection,  of  an  infinite  number  of  values. 

Definition, — A  Constant  is  a  quantity  susceptible  of 
but  one  value  in  any  given  connection. 

Remark. — In  problems  of  analysis,  constants  impose  the  con- 
ditions; variables  are  subject  to  them.  Constants  are  represented 
by  the  first  letters  of  the  alphabet;  variables  by  the  last.  At 
times,  both  are  designated  by  such  Greek  letters  as  may  be  con- 
venient. 

22.  Functions. — In  the  investigations  belonging  to 
Indeterminate  Geometry,  the  variables  are  so  connected 
by  the   conditions    of  the   problem   in   hand,   that   any 
change  in  the  value   of  one  produces   a   corresponding 
change  in  that  of  the  others. 

Definition. — A  Function  is  a  variable  so  connected 
with  others,  that  its  value,  in  every  phase  of  its  changes, 


INTE  OD  UCTION.  29 

is  derived  from  theirs  in  a  uniform  manner.  Thus,  in 
y  =  ax  -f  6,  y  is  a  function  of  x;  in  z2=mx*  -f-  ny2  -\-  I, 
z  is  a  function  of  x  and  ?/. 

Remark. — Functions  are  classed,  according  to  the  number  of 
the  variables  on  which  they  depend,  as  functions  of  one  variable, 
functions  of  two  variables,  etc. 

23.  With  these  definitions  in  view,  we  may  state  our 
Fundamental    Principle    with   greater    exactness,   thus : 
Every  equation  betiveen  variables  that  denote  the  co-ordi- 
nates of  a  pointy  represents,  in  general,  a  geometric  form. 
The  proof  of  this  now  follows. 

24.  Equations    between    Co-ordinates:    their 
Geometric  Meaning. — Every  equation  is  the  expres- 
sion of  a  constant  relation  between  the  variables  which 
enter  it.     Further,   if  we   solve    any  equation    for   one 
of  its  variables  in  terms  of  the  others,  it  becomes  ap- 
parent that   such  variable    is    a   function    of  the   rest. 
Accordingly,  by  varying  either,  we  may  cause  all   of 
them  to   vary   together,   by  differences    as   great  or  as 
small  as  we  please;   but,  so  long  as  the  constants  that 
express  the  manner  in  which  each  is  derived  from  the 
others  remain  unchanged,  all  the  changes  must  comply 
with  one  uniform  law.     That  is,  whatever  be  the  absolute 
value  of  either  variable,  its  relative  value,  as  compared 
with  the  others,  is  always  the  same.     If  either  changes 
by  infinitely  small  differences,  the  others  must  change 
by  corresponding  infinitesimals. 

If,  then,  we  assume  that  the  variables  in  an  equation 
denote  co-ordinates,  the  equation  itself  must  represent  a 
number  of  points,  as  many  as  we  please,  all  of  which 
have  co-ordinates  of  the  same  relative  values.  Now, 
since  these  co-ordinates  vary  by  differences  as  small 
as  we  please,  the  equation  really  represents  an  infinite 


30  ANALYTIC  GEOMETRY. 

number  of  points,  lying  infinitely  near  to  each  other, 
and  thus  forming  a  continuous  series.  This  continuous 
series  of  positions,  moreover,  has  a  definable  form,  of 
the  same  nature  in  all  its  parts;  since,  from  the 
definition  of  an  equation,  every  point  in  the  infinite 
succession  must  comply  with  a  law  of  position,  the  same 
for  all:  —  a  law  expressed  by  the  constants  in  the  equa- 
tion, which  subject  the  variable  co-ordinates  to  an  inflex- 
ible relation  in  value.  Every  equation  between  variables 
that  denote  co-ordinates  must  therefore,  in  general, 
represent  a  geometric  form. 

Remark. — It  will  represent  a  line  or  a  surface,  according  as 
the  co-ordinates  are  taken  in  a  plane  or  in  space. 

25.  A  few  illustrations  will  render  the  principle  just 
proved  still  more  apparent.  For  the  sake  of  variety,  we 
will  take  these  from  the  converse  point  of  view,  from 
which  it  will  appear  that  every  attempt  to  state  a  law  of 
form  in  algebraic  symbols  results  in  an  equation  between 
co-ordinates.  To  simplify,  let  us  confine  ourselves  to 
rectilinear  co-ordinates  in  a  given  plane,  and  (since  the 
axes  of  reference  may  be  any  two  intersecting  right 
lines)  suppose  XX,  Y'Y  to  intersect  at  right  angles: 
the  co-ordinates  OM,  MP  will  then  be  at  right  angles 
to  each  other. 

First:  Let  it  be  required  to 
represent  in  algebraic  symbols 
a  right  line  parallel  to  the  axis 
Y'Y.  The  law  of  this  form  -^— 
plainly  permits  the  variable 
point  P  of  the  line  to  be  at  any 


IP 


distance  above  or  below  X'X, 
but  restricts  it  to  being  at  a 
constant  distance  from  Y'Y:  a  condition  imposed  by 


TR  OD  UCTION. 


31 


assuming  that  y  =  MP  varies  without  limit,  while,  at 
the  same  time,  x  =  OM  remains  unchanged.  Thus  we 
see  that,  in  a  right  line  parallel  to  the  axis  Y'Y9  the 
co-ordinate  y  has  no  determinate  value,  but  the  co- 
ordinate x  has  a  fixed  and  unchangeable  value.  Hence 
the  algebraic  expression  for  such  a  line  is  the  equation 

x  =  constant. 

Similarly,  a  right  line  parallel  to  the  axis  X'X  is  rep- 
resented by  the  equation 

y  =  constant. 

Second:  Let  it  be  required  to  represent  algebraically 
a  circle  whose  center  is  at  the  origin  0.  The  law  of 
this  form  is,  that  the  variable 
point  P  shall  maintain  a  con- 
stant distance  from  0.  But  it 
is  obvious,  upon  inspecting  the 
diagram,  that  the  distance  of  any 
point  from  the  origin  is  equal 
to  V  x2-  -\-  y2.  Hence,  the  con- 
dition that  the  point  shall  be 
upon  a  circle  whose  center  is 

at  0,  gives  us  V  x2  +  f  --  constant  ;  and,  squaring,  we 
represent  the  circle  by  the  equation 

X2  _j_  yi  __  constant  =  r-. 

26.  Let  us  now  return  to  a  more  exact  consideration 
of  the  Fundamental  Principle.  The  student  will  have 
noticed,  that,  in  both  forms  of  stating  it  hitherto,  —  the 
forms  used  in  Arts.  23  and  24,  —  we  have  been  careful 
to  say  that  it  is  true  in  general.  This  restriction  is  nec- 
essary, and  of  great  importance  ;  for  there  are  certain 
An.  Ge.  6. 


32  ANALYTIC  GEOMETRY. 

equations  which  no  real  values  of  the  variables  will  sat- 
isfy :  and  such,  of  course,  can  denote  only  imaginary,  or 
impossible,  forms.  Others  can  only  be  satisfied  by  in- 
finite values  of  the  variables,  and  consequently  denote  a 
series  of  points  situated  at  infinity :  a  conception  as  im- 
possible, geometrically,  as  that  corresponding  to  the 
previous  class  of  equations.  Others,  again,  can  be  sat- 
isfied by  only  one  set  of  real  values  for  the  variables, 
and  therefore  represent  a  single  point ;  while  others, 
which  can  be  satisfied  by  a  fixed,  finite  number  of 
values,  but  by  no  others,  represent  a  finite  number  of 
separate  points.  Others,  still,  are  satisfied  by  distinct 
sets  of  values,  each  set  being  capable  of  an  infinite 
number  of  values  within  itself,  and  having  a  distinct 
relation  among  the  variables  which  belong  to  it ;  and 
such  equations  represent  a  group  of  distinct,  though 
related  forms. 

All  this  makes  it  clear  that,  to  hold  universally,  our 
Fundamental  Principle  must  be  stated  in  more  abstract 
terms.  We  should  be  obliged  to  say,  merely,  that  every 
equation  between  co-ordinates  represents  some  conception 
relating  to  form  or  position,  were  it  not  that  the  happy 
expedient  of  a  technical  term  saves  us  from  this  cum- 
brous circumlocution.  It  being  established  that  every 
equation  between  co-ordinates  has  some  equivalent  in  the 
province  of  geometry,  it  only  remains  to  assign  a  name 
to  that  equivalent  —  a  name  generic  enough  to  include 
not  only  surfaces  and  lines,  but  all  the  exceptional 
cases,  real,  imaginary,  or  at  infinity,  that  have  been 
mentioned  above. 

27.  Loci. — To  include  all  the  cases  that  may  arise 
under  the  conception  that  an  equation  has  geometric 
meaning,  the  term  locus  is  used. 


INTR  OD  UCTION.  33 

Definition. — A  Locus  is  the  series  of  positions,  real  or 
imaginary,  to  which  a  point  is  restricted  by  given  con- 
ditions of  form. 

Corollary. — Since  the  locus  is  the  geometric  equiva- 
lent of  the  equation,  we  may  state  the  Fundamental 
Principle  of  Indeterminate  Geometry  universally  as  fol- 
lows :  Every  equation  between  variables  which  denote  the 
co-ordinates  of  a  point.,  represents  a  locus. 

28.  The  locus  being  the  fundamental  conception  of 
purely  analytic  geometry,  it  is  of  the  utmost  importance 
that  correct  views  of  it  be  secured  at  the  outset.  The 
beginner  is  liable  to  conceive  of  it  loosely,  or  else  too 
narrowly.  To  guard  against  these  errors,  let  us  illus- 
trate what  has  been  said  or  implied  above  somewhat 
more  at  length. 

I.  Classification. — Loci    are    either    Geometric    or 
merely  Analytic :  the  former,  when  they  can  be  repre- 
sented in  a  diagram ;   the  latter,  when  they  can  not. 

Creometric  Loci  include  real  surfaces,  lines,  points,  and 
related  groups  of  either. 

Merely  Analytic  Loci  include  imaginary  loci  and 
loci  at  infinity,  and  loci  to  be  explained  hereafter  under 
the  conception  of  a  locus  in  general.  The  first  have  no 
existence  whatever,  except  in  the  equations  which  sym- 
bolize them;  the  last  two  exist  to  abstract  thought,  but 
can  neither  be  drawn  nor  imagined.  The  value  of  con- 
sidering these  merely  analytic  loci,  lies  in  their  important 
bearings  upon  some  of  the  higher  problems  of  the  science. 

II.  Conformity  to  Law. — It  is  essential  to  the  con- 
ception of  a  locus,  that  it  shall  conform  to  some  definite 
law.     No  form  that  comes  within  the  scope  of  analytic 
geometry  can  be  generated  at  hazard;   no  locus  is  the 
least  capricious.     For  it  is  always  the  counterpart  of  an 


34 


ANALYTIC  GEOMETRY. 


Y' 


equation ;  and  every  equation,  by  means  of  its  constants, 
maintains  among  its  variables,  throughout  their  infinite 
changes,  a  uniform  relation  in  value.  We  must  there- 
fore avoid  the  error  of  supposing  that  a  broken,  irregular, 
mixed  figure,  such  as  the  line  in  the  annexed  diagram, 
or  such  as  we  dash  off  with  the 
hand  at  a  scribble,  is  a  locus.  Y 

On  the  contrary,  a  locus  is,  in  a 
certain  important  sense,  homo- 
geneous. That  is,  throughout  its 
whole  extent,  it  is  so  far  alike  x- 
as  to  be  represented  by  one 
equation,  and  but  one.  No 
point  in  it  can  be  found  whose 
co-ordinates  do  not  satisfy  this 
one  equation. 

HI.  Variety  of  Meaning. — The  idea  of  the  locus 
should  be  conceived  broadly  enough  to  include,  in  addi- 
tion to  surfaces  and  lines,  the  various  exceptional  species 
enumerated  in  Art.  26.  The  attention  has,  perhaps,  been 
sufficiently  called  to  cases  where  it  is  a  point,  or  a  series 
of  separate  points,  and  where  it  is  imaginary,  or  at  infinity. 
But  it  is  worth  while  to  repeat  that  a  locus  is  not  neces- 
sarily a  single  figure.  For  example,  the  equation 


represents,  as  will  be  proved 
in  the  treatise  which  is  to  fol- 
low, two  right  lines,  such  as 
AA',  BBf,  bisecting  the  sup- 
plemental angles  between  the  X' 
axes.  Again,  as  will  also  be- 
come evident  upon  a  further 
acquaintance  with  the  subject, 
the  equation 


INTRODUCTION. 


35 


(x2  +  y2  —  r-)  (x2  +  if—rf 

represents  three  circles,  such 
as  My  M'7  and  M"9  having 
a  common  center  at  the 
origin. 

Examples  of  this  kind 
might  be  greatly  multi- 
plied, but  these  are  perhaps 
enough  to  render  the  prin- 
ciple clear,  and  to  fix  it  in  the  memory. 


II.    THE   METHOD    OUTLINED  ;    IN   WHAT    SENSE    IT   IS 
ANALYTIC. 

29.  The  following  is  a  very  simple  example  of  the 
method  by  which  Analytic  Geometry  investigates  the 
properties  of  figures.  The 
beginner,  of  course,  must 
accept  upon  authority  the 
meaning  of  the  equations 
employed. 

To  prove  that  the  tangent 
to  a  circle  is  perpendicular  to 
the  radius  drawn  to  the  point 
of  contact. — Let  the  axes  be 

rectangular,  and  the  center  of  the  circle  at  the  origin. 
The  equation  to  the  circle  is,  in  that  case, 

/y-2        I        0.2    ~,2 

'    U    ~ 

The  equation  to  the  tangent  at  any  point  P,  whose  co- 
ordinates are  x\  y' ,  is 


x'x  -f-  y'y  —  r2 


36  ANALYTIC   GEOMETRY, 

The  equation  to  the  .radius  OP,  referred  to  the  same 
axes,  is 

xfy  —  y'x  =  0  (2). 

Now,  it  is  known  that  when  two  equations  of  the  first 
degree,  referred  to  rectangular  axes,  interchange  the 
co-efficients  of  x  and  ?/,  at  the  same  time  changing  the 
sign  of  one  of  them,  they  represent  two  right  lines 
mutually  perpendicular.  Inspecting  (1)  and  (2),  we  see 
that  they  answer  to  this  condition.  Hence,  the  lines 
which  they  represent  are  mutually  perpendicular. 

30.  Generalizing  from  the  foregoing  illustration,  we 
may  sum  up  the  method  of  our  science  as  follows : 

I.  Any  locus,  the  subject  of  investigation,  is  repre- 
sented by  its  equation. 

II.  This   equation  is  then   subjected   to    such   trans- 
formations,   or   such    combinations    with    the    equations 
to   other  loci,  as  the    conditions   of   the   problem   may 
require. 

III.  The  geometric  meaning  of  these  transformations 
and   combinations,  as   derived   from   the    convention   of 
co-ordinates,  is  duly  noted.     In  the  same  way,  the  form 
of  the  final  result  is   interpreted.     Thus  the  properties 
of  the  locus  are  deduced  from  the  mere  form  of  its  equa- 
tion. 

31.  We   have    now  reached  a   point  from  which  to 
obtain  a  clear  view  of  the  reasons  why  geometry,  when 
treated  by  means  of  algebra,  should  be  called  analytic. 
We  must,  in  the  first  place,  warn  the  beginner  that  the 
reason   most   obviously   suggested   by  the   method  just 
described,  is  not  among  them.     It  is  true,  certainly,  that 
this  method  assumes  the  equation  to  any  locus  to  be  the 
synthesis,  or  expression  in   a  single   formula,  of  all  its 


1NTR  OD  UCTION.  3? 

properties.  It  is  true  that  these  properties  are  drawn 
out  from  the  equation  by  a  process  of  real  analysis  — 
namely,  by  solving  or  transforming  the  equation,  thus 
causing  it  to  give  out  the  several  conditions  which  its 
original  form  unites  in  one  symbol. 

But  this  obvious  fact,  that  we  proceed  from  a  com- 
plex unity  to  the  elements  that  have  vanished  into  it, 
by  directly  taking  apart  the  unity  itself,  is  not  the  dis- 
tinctive reason  for  calling  the  Geometry  of  the  Equation 
analytic ;  for  it  is  a  fact  which  does  not  distinguish  it 
from  the  Geometry  of  the  Diagram.  In  this  last-named 
form  of  the  science,  the  whole  scheme  of  demonstration 
consists  merely  in  developing  what  certain  definitions 
and  axioms  imply :  that  is,  the  basis  of  the  reasoning1 — 
all  that  gives  it  force  and  validity  —  is  analytic.  And, 
in  fact,  the  same  is  true  in  every  department  of  math- 
ematics. 

32.  In   short,  the   term    analytic   is   applied   to   the 
Geometry  of  the  Equation,  not  so  much  by  way  of  con- 
trast, as  of  emphasis.     It  should  not  be  taken  as  imply- 
ing that  the  Geometry  of  the  Diagram  is  wholly  synthetic, 
and  the  Geometry  of  the  Equation  wholly  analytic,  each 
to  the  exclusion  of  the  other.     Both  are  analytic,  both 
synthetic ;   and  in  both,  the  vital  principle  of  the  proofs 
is  analysis.     But  to  the  Geometry  of  the  Equation,  the 
character  of  analysis  belongs  in  a  special  sense,  and  in  a 
higher  degree;  just  as  that  of  synthesis  belongs,  in  a 
special  sense  and  higher  degree,  to  the  Geometry  of  the 
Diagram.     It  involves,  moreover,  certain  phases  of  an- 
alysis, of  a  higher  and  more  subtle   kind   than  ordinary 
geometry  attains. 

33.  Now,  this  special  analytic  character,  it  owes  to 
its  use  of  the  algebraic  symbol.     The  question  therefore 


38  ANALYTIC   GEOMETRY. 

naturally  arises  :  How  does  the  use  of  this  symbol  bring 
with  it  this  special  character  ?  The  answer  is  —  In  two 
ways :  first,  by  giving  scope  to  the  analytic  tendency 
inseparable  from  algebraic  investigation ;  secondly,  by 
introducing  the  convention  of  co-ordinates,  with  its 
added  elements  of  analysis.  We  will  illustrate  both  of 
these  ways  somewhat  in  detail. 

34.  Special  Analytic  Character  of  the  Alge- 
braic Calculus. — This  is  due  to  two  facts :  the  first, 
that  operations  with  symbols  necessarily  thrust  into  prom- 
inence the  analytic  phase  of  the  thinking  they  imply, 
while  they  obscure  the  synthetic ;  the  second,  that  the 
Theory  of  Equations  —  the  essence  of  the  science  of 
algebra,  and  the  ground  upon  which  all  its  investiga- 
tions are  based  —  is  an  application  of  analysis,  pecu- 
liarly complex  and  subtle.  That  these  are  facts,  will 
best  be  seen  by  considering  them  separately. 

I.  To  exhibit  the  first,  let  it  be  borne  in  mind  that 
every  demonstration  involves  both  analysis  and  syn- 
thesis—  analysis,  in  thinking  out  the  steps  connecting 
the  premise  with  the  conclusion;  synthesis,  in  arrang- 
ing those  steps  in  their  due  order,  and  constructing  the 
conclusion  as  their  unity.  Now,  if  we  use  ordinary 
language  in  making  this  array,  clearness  can  not  be 
secured  without  stating  these  steps  one  by  one ;  thus  we 
seem  to  begin  with  parts,  and  to  construct  the  conclusion 
as  the  whole  which  they  compose. 

But  if  we  employ  algebraic  language,  our  premise  is 
written  down  in  a  formula,  and,  at  the  outset,  our  atten- 
tion is  fixed  upon  it  as  a  whole.  The  formula  is  the 
permanent  object  of  our  thought;  the  operations,  the 
transformations  it  undergoes,  seem  transient  and  subor- 
dinate, and  their  results  but  dependent  phases  of  its 


INTE  OD  UCTION.  39 

original  form.  Derived  from  the  formula  by  a  partic- 
ular series  of  transformations,  while  a  number  of  others 
are  equally  possible,  the  conclusion  stands  in  the  mind, 
not  as  a  whole  but  rather  as  a  part.  It  appears  to  us 
as  but  one  of  many  elements  involved  in  the  original 
formula  —  elements  that  may  be  made  to  show  them- 
selves, if  we  apply  other  transformations.  Thus  the 
synthetic  phase,  though  as  real  here  as  in  using  ordi- 
nary language,  is  lost  to  view,  and  we  are  only  con- 
scious of  the  analytic. 

II.  That  the  Theory  of  Equations  involves  a  subtle 
and  peculiar  form  of  analysis,  is  obvious,  and  need  not 
be  enlarged  upon.  It  is  sufficient  merely  to  recall  its 
topics  and  their  accessories,  such  as  the  Doctrine  of 
Co-efficients,  the  Theory  of  Roots  —  their  Number, 
Form,  Situation,  and  Limits,  the.  Discussion  of  Series, 
and  the  Binomial  Theorem.  But  it  is  important  to 
mention,  that,  so  controlling  a  part  does  this  Theory 
play  in  the  whole  science  of  algebra,  and  so  emphat- 
ically does  it  embody  a  method  peculiarly  analytic,  the 
science  itself,  from  its  earliest  years,  has  been  known  by 
the  name  of  Analysis.  And  it  was  mainly  in  allusion  to 
the  fact  that  the  Geometry  of  the  Equation  brings  the 
discussion  of  Form  within  the  scope  of  this  Theory,  that 
the  title  analytic  was  originally  applied  to  it. 

35.  Elements  of  Analysis  added  by  the  Con- 
vention of  Co-ordinates. — This  convention  has  a 
twofold  analytic  meaning: 

I.  First,  it  asserts  that  the  conceptions  of  Position 
and  Form  are  merely  relative  ones,  always  implying 
certain  fixed  limits  to  which  they  are  referred  —  the 
positions  of  points  are  their  distances  from  these  limits ; 
the  forms  of  loci  are  the  relative  positions  of  their  con- 
An.  Ge.  7. 


40  ANALYTIC   GEOMETRY. 

stituent  points.  This  assertion  reaches  the  real  essence 
of  Position  and  Form,  and  gives  to  the  science  based 
upon  it  an  element  of  analysis  not  attained  in  the  Ge- 
ometry of  the  Diagram. 

II.  Secondly,  in  referring  the  form  of  every  locus  to 
fixed  limits  by  means  of  the  co-ordinates  of  every  point, 
the  convention  really  determines  that  form  by  decom- 
posing it  into  infinitely  small  elements.  It  thus  brings 
the  form  under  the  highest  analytic  conception  known 
to  mathematics,  and  prepares  for  its  discussion  by  the 
various  branches  of  the  Infinitesimal  Calculus. 

3G.  To  recapitulate :  The  Geometry  of  the  Equation 
is  called  Analytic,  first,  because  its  use  of  algebraic 
processes  puts  forward  the  analytic,  and  retires  the 
synthetic  phase  of  every  demonstration,  thus  rendering 
us  conscious  of  investigation  rather  than  of  proof; 
secondly,  because  its  method  consists  in  applying  those 
special  modes  of  analysis  which  mark  the  Theory  of 
Equations;  thirdly,  because  its  convention  of  co-ordi- 
nates penetrates  to  the  real  nature  of  Position  and 
Form,  resolving  them  into  their  essential  constituents  — 
Fixed  Limits  and  Distance ;  finally  and  most  signifi- 
cantly, because  it  resolves  all  Forms  into  elements 
infinitely  small,  and  thus  brings  the  discussion  of  loci 
within  the  sphere  of  the  Infinitesimal  Calculus  —  the 
highest  expression  of  mathematical  analysis. 

37.  These  facts  constitute  a  sufficient  reason  for  pre- 
ferring to  call  the  science  Analytic  Geometry,  rather 
than  Algebraic.  The  former  title,  more  forcibly  than 
the  latter,  calls  up  the  characteristics  of  its  method,  as 
they  have  been  detailed  above.  Moreover,  the  term 
algebraic  is  ambiguous;  for  it  is  generally  used,  in  con- 
trast to  transcendental,  to  characterize  operations  which 


INTR  OD  UCTION.  41 

involve  only  addition,  subtraction,  multiplication,  di- 
vision, or  involution  and  evolution  with  constant  indices. 
But  Analytic  Geometry  considers  all  loci  whatsoever, 
not  only  Algebraic  but  Transcendental,  whether  the 
latter  be  Exponential,  Logarithmic,  or  Trigonometric. 

38.  The  superiority  of  the  Geometry  of  the  Equation 
to   that  of  the  Diagram   consists  partly  in   its   greater 
brevity  and  elegance,  but  mainly  in  its  greater  power 
of  generalization.     For  since  the  equation  to  any  locus 
is  the  complete  synthesis  of  all  its  properties,  our  power 
of  investigating  and  discovering  these  is  limited  only  by 
our  ability  to  transform  the  equation  and  to  determine 
the  form,  limits,  number,  and  situation  of  its  roots.    And 
since  every  equation  denotes  some  locus,  the  equations 
of  the  several  degrees  may  be  discussed  in  their  most 
general  forms.     Loci  may  thus  be  grouped  into  Orders, 
according  to  the  degree  of  their  equations,  and  proper- 
ties common  to  an  entire  Order  may  be  discovered  by 
absolute  deduction  —  properties  which,  if  they  could  be 
established    at    all    by    ordinary    geometry,   would    in- 
volve  the   most   tedious   processes    of   comparison   and 
induction. 

III.    THE    SUBDIVISIONS    OF    THE   SCIENCE. 

39.  The  subdivisions  of  Indeterminate  Geometry  refer 
to  the  nature  of  the  loci  discussed  in  each ;  and  these  are 
classified,  primarily,  according  to  the  form  of  their  equa- 
tions ;  secondarily,  according  to  their  situation  in  a  plane, 
or  in  space. 

40.  Hence,  the  first  division  of  Indeterminate  Geom- 
etry is  into  TRANSCENDENTAL  and  ALGEBRAIC. 

Transcendental    Geometry    discusses    those    loci 
whose  equations  involve  transcendental  functions ;  that  is, 


42  ANALYTIC  GEOMETRY. 

functions  which  depend  on  either  a  variable  exponent,  a 
logarithm,  or  one  of  the  expressions  sin,  cos,  tan,  etc. 

Algebraic  Geometry,  those  whose  equations  in- 
volve none  but  algebraic  functions;  that  is,  functions 
which  imply  only  the  operations  of  addition,  subtraction, 
multiplication,  division,  or  involution  and  evolution  with 
constant  indices. 

41.  Algebraic  loci  are  classed  into  Orders,  according 
to  the  degree  of  their  equations  referred  to  rectilinear 
axes.     Thus,  the  locus  whose   equation  is   of  the  first 
degree,  is   called  the  locus   of  the   First  order;    those 
whose  equations  are  of  the  second  degree  are  called  loci 
of  the  Second  order ;  and  so  on. 

The  loci  of  the  First  and  Second  orders,  on  account 
of  their  simplicity,  symmetry,  and  limited  number,  are 
considered  to  form  a  class  by  themselves ;  and  those  of 
all  higher  orders  are  grouped  together  as  a  second  class. 

42.  Accordingly,  the  second  division  of  our  subject 
is  that  of  Algebraic  Geometry  into  ELEMENTARY  and 
HIGHER. 

Elementary  Geometry  is  the  doctrine  of  loci  of 
the  First  and  Second  orders. 

Higher  Geometry  is  the  doctrine  of  loci  of  higher 
orders  than  the  Second. 

43.  Each  of  these  divisions  based  upon  the  form  of 
the   equations    considered,    falls    into    the    province    of 
Plane   or  of  Solid  Geometry,   according   as   the   equa- 
tions are  between  plane  co-ordinates,  or  co-ordinates  in 
space;    that   is,  according   as   the   system   of  reference 
is  two  intersecting  lines,  giving  rise  to  two  co-ordinates 
for  every  point ;  or  three  intersecting  planes,  giving  rise 
to  three  co-ordinates. 


INTRODUCTION. 


43 


Hence,  we  have  GEOMETRY  OF  Two  DIMENSIONS,  and 
GEOMETRY  OF  THREE  DIMENSIONS. 

44.  The  relations  which  the  various  divisions  of  An- 
alytic Geometry  sustain  to  each  other,  will  be  best 
understood  from  the  following  • 


SYNOPTICAL   TABLE   OF    DIVISIONS. 

'  DETERMINATE. 


ANALYTIC 
GEOMETRY 


INDETERMINATE  - 


f  OF    TWO   DIMENSIONS. 
TRANSCENDENTAL  J 

V.  OF  THREE  DIMENSION& 


ALGEBRAIC  - 


HIGHER 


ELEMENTARY 


OF    TWO 
DIMENSIONS. 


OF    THREE 
DIMENSIONS. 


OF   TWO 
DIMENSIONS. 


OF  THREE 
DIMENSIONS. 


45.  In  the  present  treatise,  we  do  not  purpose  more 
than  an  introduction  into  the  wide  domain  which  the 
foregoing  scheme  presents.  We  shall  confine  ourselves 
to  Elementary  Geometry,  not  entering  upon  the  other 
departments  any  further  than  may  prove  necessary  in 
order  to  present  the  subject  in  its  true  bearings.  And, 
restricting  ourselves  to  the  discussion  of  loci  of  the  First 
and  Second  orders,  we  can  not  within  that  compass 
give  more  than  a  sketch  of  the  doctrine,  methods, 
and  resources  of  the  science,  in  its  present  advanced 
condition.  We  shall,  however,  consider  the  loci  of  the 


44 


ANALYTIC  GEOMETRY. 


first  two  orders,  both  in  a  plane  and  in  space.  Ac- 
cordingly, our  treatise  falls  naturally  into  two  Books: 
the  first,  upon  Plane  Co-ordinates;  the  second,  upon 
Co-ordinates  in  Space. 


NOTE. 

As  Greek  characters  are  extensively  used  in  all  analytic  investi- 
gations, and  as  we  shall  very  frequently  employ  them  in  the  following 
pages,  we  subjoin  a  list  for  the  benefit  of  readers  unacquainted  with 
Greek. 


A  a 
B/3 
Ty 
A  6 
E  e 
Z  C 
H  r, 
0  0 


alpha. 

beta. 

gamma. 

delta. 

epsilon. 

zeta. 

eta. 

theta. 


I  i 
K  K 

A  A 
MyU 

N  v 
Hf 
0  o 


iota. 

kappa. 

lambda. 

mu. 

nu. 

«. 

omicron. 

pi. 


Pp 

2  a 
T  r 
T  v 


rho. 

sigma. 

tau. 

upsilon. 

phi. 

chi. 

2)si. 

omega. 


BOOK  FIRST: 
PLANE  CO-ORDINATES. 


PLANE  CO-ORDINATES. 


PART  I. 

THE  REPRESENTATION  OF  FORM  BY 
ANALYTIC  SYMBOLS. 

46.  In    applying  to  plane   curves   of  the  First   and 
Second    orders   the   method  sketched  in  the   foregoing 
pages,  our  work   will   naturally  divide    itself  into    two 
portions:  we  shall  first  have  to  determine  the  equations 
which  represent  the  several  lines  to  be  discussed;  and 
then  deduce  from  these  equations  the  various  properties 
of  the  corresponding  lines.    Accordingly,  our  First  Book 
falls  into  two  parts : 

PART  I. — ON  THE  REPRESENTATION  OF  FORM  BY  AN- 
ALYTIC SYMBOLS. 

PART  II. — ON  THE  PROPERTIES  OF  CONICS. 

47.  We  say  Properties  of  Conies,  because,  as  will  be 
shown  hereafter,  all  the  lines  of  the  First  and  Second 
orders  may  be   formed  by  passing   a  plane   through   a 
right  cone  on  a  circular  base.     By  varying  the  position 
of  the  cutting  plane,  its  sections  with  the  conic  surface 

(47) 


48  ANALYTIC  GEOMETRY. 

will  assume  the  forms  of  the  several  lines;  and  these 
may  therefore  be  conveniently  grouped  under  the  gen- 
eral name  of  Conic  Sections,  or  Coriics. 

It  must  be  added,  however,  that  this  use  of  the  term  Conies  is 
wider  than  ordinary.  For,  speaking  strictly,  we  mean  by  the 
Conies  the  curves  of  the  Second  order  alone ;  namely,  the  Ellipse, 
the  Hyperbola,  and  the  Parabola;  and  ordinarily  the  line  of  the 
First  order  is  not  included  in  the  term. 

48.  We  shall  therefore  proceed  to  develop  the  modes 
of  representing  in  algebraic  language  the  Right  Line, 
the  Circle,  the  Ellipse,  the  Hyperbola,  and  the  Parabola. 
And  as  these  modes  of  representation  are  all  derived 
from  the  conventions  adopted  for  representing  a  point, 
we  shall  begin  by  explaining  in  full  the  principal  forms 
of  those  conventions,  which  were  merely  sketched  in 
the  Introduction. 

We  shall  obtain,  first,  the  formulae  in  ordinary  use,  or 
those  which  may  be  said  to  constitute  the  Older  Geom- 
etry; and,  afterward,  those  belonging  to  the  Modern 
Geometry,  based  on  what  is  called  the  Abridged  No- 
tation. 


CHAPTER   FIRST. 

THE  OLDER  GEOMETRY :    BILINEAR  AND  POLAR 
CO-ORDINATES. 

SECTION   I.  —  THE   POINT. 

BILINEAR  OR  CARTESIAN  SYSTEM  OF  CO-ORDINATES. 

4O.  Resuming    the    topic    and    diagram    of  Art.    19, 
let  us  examine   the   Cartesian*   system  of  co-ordinates 


*  Cartesian,  from  Cartesius,  the  latinized  form  of  Descartes'  name. 


BILINEAR  CO-ORDINATES.  49 

more  minutely,   and   develop   its    elements   in   complete 
detail. 

P  being  any  point  on  a  given 
plane,  two  right  lines  X'X  and 
Yr  Y  are  drawn  in  that  plane, 
intersecting  each  other  in  0. 
From  P,  a  line  PM  is  drawn 
parallel  to  Y'Y. 

The  distances  OM,  MP  being 
known,  the  position  of  P  is  de- 
termined. These  distances  are  called  the  bilinear  co- 
ordinates of  the  point  P.  They  are  also  termed  recti- 
linear, and  sometimes  parallel,  co-ordinates.  They  are 
frequently  cited  as  the  Cartesian  co-ordinates  of  the 
point. 

The  co-ordinate  MP,  drawn  parallel  to  Y'Y,  is  called 
the  ordinate  of  the  point  P.  The  co-ordinate  OM, 
which  the  former  cuts  off  from  X'X,  is  called  the  abscissa 
of  the  point.  The  abscissa  of  a  point  is  represented  by 
the  symbol  x;  its  ordinate,  by  the  symbol  y. 

The  two  lines  X'X  and  Y'Y  are  called  the  axes  of 
reference,  or  simply  the  axes.  X'X,  on  which  the  ab- 
scissas are  measured,  is  called  the  axis  of  abscissas;  or, 
more  briefly,  the  axis  of  x.  Y'  Y,  parallel  to  which  the 
ordinates  are  drawn,  is  called  the  axis  of  ordinates;  or, 
for  brevity,  the  axis  of  y. 

The  point  0,  in  which  the  axis  of  x  cuts  the  axis  of  y, 
is  called  the  origin. 

The  angle  YOX  is  called. the  inclination  of  the  axes. 
It  is  designated  by  the  Greek  letter  co,  and  may  have 
any  value  from  0  to  180°.  If  w  =  90°,  the  axes  and 
co-ordinates  are  said  to  be  rectangular;  if  a>  has  any 
other  value,  they  are  said  to  be  oblique. 

The   two    axes,   being   of   infinite   length,   divide   the 


50  ANALYTIC  GEOMETRY. 

whole  planar  space  about  0  into  four  angles.  These 
are  numbered  to  the  left,  beginning  at  the  line  OX. 
XOY  is  the  first  angle;  YOX'  is  the  second;  XOY', 
the  third;  and  Y'OX,  the  fourth. 

The  signs  +  and  —  are  used  in  connection  with  the 
co-ordinates,  and  are  taken  to  signify  measurement  in- 
opposite  directions.  Positive  abscissas  are  measured  to 
the  right  from  0,  as  OM;  negative  ones,  to  the  left;  as 
OM.  Positive  ordinates  are  measured  upward  from  X X, 
as  MP ;  negative  ones,  downward;  as  MP'" . 

By  attributing  proper  values  to  the  co-ordinates  x  and 
y^  and  taking  account  of  their  signs,  we  may  represent 
any  point  in  either  of  the  four  angles.  Thus, 

>  denotes  a  point  in  the  first  angle. 
~a\        "  "  "       second    " 

~a\        "  «  "       third       " 

—  —  b  ) 


x  =  — 

y 

X  =  ~~ 


=  ~~  ^  a  \       "  "  "        fourth    " 

=  —  o) 

Corollary  1.  —  For  any  point  on  the  axis  of  x,  we  shall 
evidently  have 


while  x,  being  susceptible  of  any  value  whatever,  is  in- 
determinate. Hence,  the  equation  just  written  is  the 
equation  to  the  axis  of  x. 

Corollary  2.  —  For  any  point  on  the  axis  of  y,  we  shall 
have 

*=o, 

while  y  is  indeterminate.  Hence,  the  equation  last  writ- 
ten is  the  equation  to  the  axis  of  y. 


BILINEAR  CO-ORDINATES.  51 

Corollary  3. — For  the  origin,  we  shall  obviously  have 

jp  =  0, 

0  =  0; 

and  these  expressions  are  therefore  the  symbol  of  the  origin. 

Remark  1, — For  the  sake  of  brevity,  any  point 
designated  by  Cartesian  co-ordinates  is  written  and 
cited  as  the  point  x  y,  the  point  a  b,  the  point  (3,  5),  etc. 
These  expressions  are  not  to  be  confounded  with  alge- 
braic products. 

Remark  2, — The  symbols  x  and  y  are  used  for  co- 
ordinates of  a  variable  point,  and  are  therefore  general  in 
their  signification.  But  it  is  often  convenient  to  repre- 
sent particular,  or  fixed,  points  by  the  variable  symbols ; 
especially  when  their  positions,  though  fixed,  are  arbi- 
trary. In  such  cases  accents,  or  else  inferiors,  are  used 
with  the  x  and  y.  Thus  x'  yf,  x"  y",  xl  y},  x2  y2,  all  rep- 
resent points  which  are  to  be  considered  as  fixed,  but 
fixed  in  positions  chosen  at  pleasure.  This  distinction 
between  the  point  x  y  as  general,  and  points  such  as 
x'  y',  x2  y2  as  particular,  should  be  carefully  remembered. 

Remark  3, — A  point  is  said  to  be  given  by  its  co- 
ordinates, when  their  values  and  that  of  the  angle  co  are 
known.  And  when  so  given,  the  point  may  always  be 
represented  in  position  to  the  eye.  For  we  have  only  to 
draw  a  pair  of  axes  with  the  given  inclination,  and  lay 
off  by  any  scale  of  equal  parts  we  please  the  given  ab- 
scissa and  ordinate.  In  practice,  it  is  most  convenient 
to  lay  off  the  co-ordinates  on  the  axes,  and  draw  through 
the  points  thus  determined,  lines  parallel  to  the  axes ; 
the  intersection  of  the  latter  will  be  the  point  required. 
The  truth  of  this  will  be  apparent  on  inspecting  the  dia- 
gram at  the  head  of  this  article. 


52  ANALYTIC  GEOMETRY. 

EXAMPLES. 

1.  Represent  the  point  (—  5,  3)  in  rectangular  co-ordinates. 

2.  Represent  the  point  ( —  3,  —  7),  axes  oblique  and  w  =  60°. 

3.  With  the  same  axes  as  in  Ex.  2,  represent  the  points  (1,  2), 
(-3,4),  (-5,  -6),  (7,  -8). 

4.  Represent  the  points  corresponding  to  the  co-ordinates  given 
in  Ex.  3,  axes  being  rectangular. 

5.  Given  w  =  135°,  to  represent  the  points  (—4,  —  1),  (8,  2) 

(2,  8)- 

6.  Given  «  =  90°,  represent  the  points  (3,  4),  (3,  —  4),  (—  3,  4), 
(-3, -4). 

7.  With  same  axes,  represent  (6,  8)  and  (8,  6) ;  also  (6,  —  8)  and 
(-8,6). 

8.  With  a  still  =  90°,  represent  the  distance  between  (2,  3) 
and  (4,  5). 

9.  With  same  axes,  represent  the  distance  between  (4,  5)  and 
(-3,2). 

10.  Axes  rectangular,  represent  the  distance  between  (0,  6)  and 

(_  5,  —  5) ; the  distance  between  (0,  0)  and  (6,  0).     Does  the 

latter  distance  depend  on  the  value  of  w,  or  not? 

POLAR    SYSTEM    OF    CO-ORDINATES. 

5O.  A  second  method  of  representing  the  position 
of  a  point  on  a  given  plane,  is  founded  on  the  fact  that 
we  naturally  determine  the  position  of  any  object  by 
finding  its  direction  and  distance  from  our  own. 

Hence,   if  we   are   given    a 
fixed  point  0  and  a  fixed  right       F 
line  OX  passing  through  it,  we  > 

shall  evidently  know  the  posi-  j 

tion  of  any  point  P,  so  soon  as 

we  have  determined  the  angle       p»      v»- ..---''      ^ 

XOP  and  the  distance  OP. 
This  method  of  representing  a 
point  is  known  as  the  method  of  Polar  Co-ordinates. 


POLAR  CO-ORDINATES.  53 

The  fixed  point  0  is  called  the  pole;  the  fixed  line  OX, 
the  initial  line. 

The  distance  OP  is  termed  the  radius  vector;  the  angle 
XOP,  the  vedorial  angle.  It  is  customary  to  represent 
the  former  by  the  letter  p,  and  the  latter  by  6.  In  this 
system,  accordingly,  a  point  is  cited  as  the  point  p  6, 
the  point  //  6',  etc. 

By  attributing  proper  values  to  p  and  #,  we  may  repre- 
sent any  point  whatever  in  the  plane  PXO.  Thus, 

'  >  denotes  the  point  P. 

6    =  XOP    ) 

P'          OF    1  (<      „      p, 

5'   =AW    j 


prrr 

'"  =  XOP"f 

The  student  will  observe  that  all  the  angles  0,  0',  6", 
6f"  are  estimated  from  XO  toward  the  left. 

Corollary  1. — For  the  pole,  we  evidently  have 

,o  =  0: 

which  may  therefore  be  considered  as  the  equation  to  that 
point. 

Corollary  2. — For  any  point  on  the  initial  line  to  the 
right  of  the  pole,  we  have 

6  =  2  n  TT  *. 

For  any  point  on  the  same  line  to  the  left  of  the  pole, 
we  have 

0  =  (2  n  +  1)  TT. 


*  We  shall  frequently  employ  the  symbol  TT  =  the  semi-circumference 
to  radius  1,  to  denote  the  angle  180° ;  on  the  principle  that  angles  are 
measured  by  the  arcs  which  subtend  them. 


54  ANALYTIC  GEOMETRY. 

In   these   expressions,  n  may  have  any  integral  value 
from  0  upward. 

Note — It  is  customary  to  measure  the  angle  6  from  XO  toward 
the  left;  and  the  radius  vector  p,  from  O  in  such  a  direction  as  to 
bound  the  angle.  When  any  distinctions  of  sign  are  admitted  in 
polar  co-ordinates,  the  directions  just  named  are  considered  positive; 
while  an  angle  measured  from  OX  toward  the  right,  and  a  radius 
vector  measured  from  O  in  the  direction  opposite  to  that  which 
bounds  its  angle,  are  considered  negative.  Thus,  the  point  P  is 
commonly  denoted  by  the  positive  angle  0  •=  XOP  and  the  positive 
vector  p=  OP ;  but  it  may  also  be  represented  by  the  negative  angle 
XOP"  =  —  {^  —  6}  and  the  then  negative  vector  p  =  OP.  And, 
again,  the  same  point  may  be  represented  by  the  positive  angle 
0"  =  6  +  TT  =  XOP"  and  the  negative  vector  p  =  OP. 

The  student  will  not  fail  to  note  that  the  signs  -f  and  — ,  as 
applied  to  a  line  revolving  about  a  fixed  point,  have  a  signification 
quite  different  from  that  in  connection  with  bilinear  co-ordinates. 
They  discriminate  between  radii  vectores,  not  necessarily  as  measured 
in  opposite  directions,  but  in  directions  having  opposite  relations  to  the 
bounding  of  the  vectorial  angle.  We  may  therefore  define  the  positive 
direction  of  a  radius  vector  to  be  that  which  extends  from  the  pole 
along  the  front  of  the  vectorial  angle;  and  the  negative,  to  be  that  ex- 
tending opposite. 

In  practice,  negative  values  of  p  and  6  are  excluded  from  the 
ordinary  formulas;  but  the  distinction  of  sign  just  explained  has  an 
important  bearing  on  the  principles  of  the  Modern  Geometry.  For 
this  reason,  it  should  be  mastered  at  the  outset. 

Remark. — To  represent  any  point  given  in  polar  co- 
ordinates, we  have  only  to  draw  the  initial  line,  and  lay 
off  at  any  point  taken  for  the  pole,  an  angle  equal  to 
the  given  angle  6  :  then  the  distance  p  being  measured 
from  the  pole,  the  required  point  is  obtained. 

EXAMPLES. 

1.  Represent  in  polar  co-ordinates  the  point  (p  =  8;  0  =  7r). 

2.  Represent  (p  =  —  8 ;  6  =  0)  and  (p  =  —  8 ;  6=ir). 

3.  Represent  (p  =  15 ;  6  =^\  and  (p  =  .5  ;   6  =  3~}  . 


DISTANCE  BETWEEN  TWO  POINTS.  55 

.  Represent  (p  =  6;    P  ==  ^H    fp  =  —  6;    0=  -  -  -g-Y  and 


5.  Represent    the    distance    between    (p  =  8;    6  =  j)     and  * 


DISTANCE    BETWEEN    ANY   TWO    POINTS. 

51.  Any  two  points  being  given  by  their  co-ordinates, 
the  distance  o  between  them  is  given.     For, 

First:    let    the    two    points   be 

x'  y'   and  x"  y".     Taking  P  and  /Y  F 

P'   to   represent   the   points,    we  /       ^^-^/ 

have 

P'P=3,  /  / 


By  Trig.,  865, 

P'P2  =  PQ2  +  PfQ2  —  2PQ.P'Q  cos  PQP 
=  PQ2  +  P'Q2  +  2PQ.P'Q  cos  YOX.    (Trig.,  825). 
That  is, 
32  =  (x"  —  x'Y+(y"  —  y')2  +  2(x"  —  x'}  (y"  —  y)cosa>. 

Corollary  1.—  If  w  =  90°,  then  (Trig.,  834)  the  last 
term  in  the  foregoing  expression  vanishes,  and  we  have, 
for  the  distance  between  two  points  in  rectangular  co- 
ordinates, 

8>  =  (x"-x'y+(y"-y<)\ 

Corollary  2.  —  For  the  distance  of  any  point  x  y  from 
the  origin,  we  have  (Art.  49,  Cor.  3) 

$2  —  #2  _j_  y2  _|_  2  xy  cos  co  : 

which,  in  rectangular  co-ordinates,  becomes 


An.  Ge.  8. 


56  ANALYTIC  GEOMETRY. 


Second:  let  the  two  points  be  p'  6'  and  p"  6".     In  the 
,  for  this  case, 

OP=p',XOP=0'; 


diagram,  for  this  case, 


Then,  as  before,  o  — x 

ppt  ==  OP2  +  P02  —  20P.  PO  cos  POP; 
that  is, 

o2  =  p'2  -f  p"'2— 2  ///>' '  cos  (0"  —  6'). 

Corollary, — For  the  distance  of  any  point  p  6  from 
the  pole,  we  have  (Art.  50,  Cor.  1) 
$2  =  p2  •  or  o  =  p  : 

which  agrees  with  our  definition  of  the  radius  vector. 

Note — In  using  the  formulae  of  this  article,  be  careful  to  observe 
the  signs  of  x/  y',  x//  y//. 

POINT   DIVIDING   IN   A   GIVEN  RATIO    THE   DISTANCE 
BETWEEN   TWO    GIVEN   POINTS. 

52.  Let  the  given  points  be  xl  yl9  x2y2 ;  and  the  given 
ratio,  m  :  n.  Denote  the  co-ordinates  of  the  required 
point  by  x  and  y. 

By  Geom.,  313,  we  have  in  the  /  j^  p 

diagram  annexed,  where  OM=  x, 
MP=y;    OM'  =  xl,  M'P  =  yj 

2}  &  2)  /O 

PR  :  RQ  ::  PP  :  PP" ; 

or,  x  —  Xt  :  x2  —  x  ::  m  :  n. 

_  mx2  -\-  nxl 

m  -f-  n 
By  like  reasoning,  we  find 

y  = 


DISTANCE  DIVIDED  IN  GIVEN  RATIO.  57 

If  the  distance  between  two  points  x±  yl9  x2  y2  were 
cut  externally  in  the  given  ratio,  we  should  have 


x  —  xl\    x  —  x2  : :  m  :  n. 
mx?>  —  nx, 


.• .     x  = 
and,  similarly, 


m  —  n 


And  this  we  should  expect  :  for,  if  the  point  P  fell 
beyond  P",  the  segment  PP"  would  be  measured  in  the 
direction  opposite  to  P'P,  and  n  would  have  the  negative 
sign. 

53.  As  the  student  may  have  surmised  from  what  he 
has  already  noticed,  formulae  referred  to  rectangular  axes 
are  generally  simpler  than  those  referred  to  oblique.  For 
this  reason,  it  is  preferable  to  use  rectangular  axes  when- 
ever it  is  practicable.  Hereafter,  then,  the  attention 
should  be  fixed  chiefly  upon  those  formulae  which  corre- 
spond to  rectangular  axes.  In  some  cases,  formulae  are 
true  for  any  value  of  co.  Such  are  those  deduced  in  the 
last  article.  In  the  examples  given  hereafter,  the  axes 
are  supposed  to  be  rectangular,  unless  the  contrary  is  men- 
tioned. 

EXAMPLES. 

1.  Draw   the  triangle    whose    vertices    are    (2,  5),    (  —  4,    1), 
(-2,  -6). 

2.  Find  the  lengths  of  the  three  sides  of  the  same  triangle. 

3.  Express  algebraically  the   condition  that  x  y  is  equidistant 
from  (2,  3)  and  (4,  5).  Ans.  x  +  y  —  1. 

4.  Find  the  distance  between  (  —  3,  0)  and  (2,  —  5). 

5.  Determine  the  co-ordinates  of  the  point  equidistant  from  the 
three  points  (1,  2),  (0,  0),  and  (—5,  -  6). 


58  ANALYTIC  GEOMETRY. 

6.  Solve  Ex.  2,  supposing  w  successively  —  and  -r  . 

o  4 

7.  Find  the  co-ordinates  of  the  point  bisecting  the   distance 
between  xi  y\  and  xz  y2 

8.  The  point  x  y  is  midway  between  (3,  4)  and  ( —  5,  —  8) : 
find  its  distance  from  the  origin, 

9.  x  y  divides  externally  the  distance  between   (2,  —  8)  and 
( —  5,  —  3)  in  the  ratio  6  :  7.     What  is  its  distance  from  the  point 
midway  between  (3,  4)  and  (6,  8)  ? 

10.  Given  the  points  (p  =  5 ;  6=  30°)  and  (p  —  6 ;  0  =  225°)  to 
find  the  distance  between  them,  and  the  polar  co-ordinates  of  its 
middle  point. 

TRANSFORMATION   OF    CO-ORDINATES. 

54.  A  point  being  given  by  its  co-ordinates,  we  can 
at  pleasure  change  either  the  axes  or  the  system  to 
which  they  refer  it.  The  process  is  called  Transforma- 
tion of  Co-ordinates. 

The  position  of  the  point  is  of  course  not  affected  by 
such  a  change.  Its  co-ordinates  merely  assume  new  values 
corresponding  to  the  new  axes  or  new  system. 

The  transformation  is  effected  by  substituting  for  the 
given  co-ordinates  their  values  in  terms  of  the  elements  be- 
longing to  the  new  limits.  General  formulae  for  these  substi- 
tutions are  easily  obtained.  In  investigating  them,  it  is 
convenient  to  consider  the  subject  in  four  cases;  namely, 

I.  To  change  the  Origin,  the  direction  of  the  axes 
remaining  the  same. 

II.  To  change  the  Inclination  of  the  Axes,  the  origin 
remaining  the  same. 

III.  To  change  System — from  Bilinears  to  Polars,  and 

conversely. 

IV.  To  change  the  Origin,  at  the  same  time  trans- 

forming by  II  or  III. 


TRANSFORMATION  OF  CO-ORDINATES. 


59 


55.  Case  First:  To  transform  to  parallel  axes  through 
a  new  origin. 

Let  x  and  y  be  the  co-ordinates  of  the  point  for  the 
primitive  axes ;  and  X  and  Y  its  co-ordinates  for  the 
proposed  parallel  axes.  Let 
m,  n  be  the  co-ordinates  of 
the  new  origin.  Then,  if  OY, 
OX  represent  the  primitive 
axes,  and  O'Y',  O'X'  the 
new :  we  shall  have  x  =  OM, 
y  =  MP;  X=0'M',  Y= 
M'P;  and  m  =  OS,  n  =  SO'. 
Now,  from  the  diagram, 

OM=OS+0'M>     and 

that  is,  x  =  m  -\-  X, 


which  are  the  required  formulae  of  transformation. 

5G.  Case  Second :  To  transform  to  new  axes  through 
the  primitive  origin,  the  inclination  being  changed. 

Let  co  represent  the  inclination  of  the  primitive  axes. 
Let  a.  --  the  angle  made  by  the  new  axis  of  x  with  the 
primitive ;  and  ft  =  that  made  by  the  new  axis  of  y. 

Drawing  the  annexed  dia- 
gram, we  have  x  =  OM,  y 
--  MP;  X  =  OM',  Y  = 
M'P;  a=XfOX  =  M'OR, 
and  ft  --  :  Y'OX  =  PM'S. 
Then,  letting  fall  PQ  and 
M'R  perpendicular  to  OJT, 
and  M'S  perpendicular  to 
PQ,  we  obtain  (Trig.,  858) 

MP  sin  PMQ  =  QP  =  RMr  +  SP. 


Q   X 


60  ANALYTIC   GEOMETRY. 

But    R M'  +  SP  =  OM'  sin  M  OR  -f  M'P  sin  PMS. 
.-.  MP  sin  PMQ  =  (W  sin  J^P  072  +  M'P  sin  P.¥'£; 

or,  y  sin  o>  =  Ar  sin  a  -\-  Y  sin  /9. 

By  dropping  perpendiculars  from  P  and  Jf' upon  OJ7, 

and  completing  the  diagram,  we  should  obtain  by  the 
same  principles 

x  sin  co  =  X  sin  (co  —  a)  -j-  Y  sin  (a>  —  ,9). 
The  details  of  the  proof  are  left  to  the  student. 

We  have,  then,  as  the  required  formulae  of  transforma- 
tion, 

x  sin  co  —  X  sin(ft>  —  «)  +  l^sin  (w  —  /9), 
?/  sin  (o  =  X  sin  «  -f-  !F"  sin  /9  : 

which  include  all  cases  of  bilinear  transformation.  The 
particular  transformations  which  may  arise  under  this 
general  case,  are  various ;  those  of  most  importance 
are  as  follows : 

Corollary  1. —  To  transform  from  rectangular  axes  to 
oblique,  the  origin  remaining  the  same.  Making  co  =  90° 
in  our  general  formulae,  we  obtain  (Trig.,  834,  841) 

x  =  X  cos  a  -\-  Y  cos  /?, 
y  —  X  sin  a  -f-  Ysin  /9. 

Or  we  may  obtain  the  formulae  geometrically,  as  follows  : 
Supposing  the  angle  YOX  to  be  a  right  angle,  OM  will 
obviously  coincide  with  OQ,  and  MP  with  QP,  and  we 
shall  have 

OM=OR  +  M'S  =  OM'  cos  M'OR  -f  M'P  cos  PM'S 

.  • .  x  =  X  cos  a  -\-  Y  cos  /9 ; 

MP  =  RM  +  SP  =  OM  sin  MOR  +  M'P  sin  PM'S 

.  • .  y  —  -X"  sin  a  -\-  Y  sin  ft. 


TRANSFORMATION  OF  CO-ORDINATES. 


61 


Corollary  2. — To  transform  from  oblique  axes  to  rectan- 
gular, the  origin  and  the  axis  of  x  remaining  the  same. 
Here  a  =  0,  and  /3  =  90°:  hence,  (Trig.,  829,  834,) 

x  sin  a)  =  X  sin  co  —  Y  cos  co, 
y  sin  co  =  Y. 

Geometrically  as  follows :  OX,  OY  being  the  primitive 
axes,  and  OX,  0 F  the  new  :  x=OM,y  =  MP; X=  OM' 
Y=MP.  By  Trig.,  859, 

OM=OM'  —  M'M 

=  OMf  —  M'P  cot  PMM' ; 

or,          x  =  X —  Ycot  co. 
. ' .  x  sin  co  =X$m  co  —  Fcos  co. 
Again,  by  Trig.,  858, 

MPsmPMM=MP; 

or,  y  sin  co  =  Y. 

Corollary  3. — To  revolve  the  rectangular  axes  through 
any  angle  6.  Here  a  =  6,  [3  =  90°  +  6,  and  co  =  90°. 
Substituting  in  the  general  formulae,  we  obtain 

x  =  X  cos  0  -f  Y  cos  (90°  +  6), 
y  =  X  sin  6  +  T  sin  (90°  +  0) : 

expressions  which,  by  Trig.,  843,  become 

x  =  X  cos  0  —  F  sin  6; 

y  =  Xsin  6  +  Fcos  6. 

Or  we  may  deduce  the  formulae 
independently,  from  the  diagram 
annexed.  Here  x  =  OM,  y  =  MP; 
X=OMr,  Y=M'P;  6  =  MfOS 
=  ROM=M'PQ.  Drawing  MS 
and  M'Q  perpendicular  respect- 
ively to  OS  and  PQ,  we  shall  have 


O    M     SX 


\ 


62  ANALYTIC  GEOMETRY. 

OM=  OS—M'Q  =  OM'  cos  M'OS—M'P  sin  M'PQ, 
MP  =  SM'  JrQP=  OM'  sin  M  '  OS  +  M'P  cos  M'PQ. 

That  is, 


M 


y=X  sin#  +  Fcosfl. 

57.  Case  Third:  To  transform  from  a  bilinear  to  a 
polar  system  of  co-ordinates,  or  conversely. 

Let  OX,  OF  be  the  primitive  rectangular  axes,  and 
OXf  the  initial  line  of  the  proposed 
polar  system.  Let  a  =  the  angle 
which  the  initial  line  makes  with  the 
axis  of  x  :  it  will  be  positive  or  neg- 
ative according  as  OX'  lies  above  or 
belotv  OX.  It  is  obvious  from  the 
diagram  that  we  shall  have 

x  =  p  cos  (6  +  a), 
y  =  p  sin  (6  -f-  a)  : 

formulae  by  which  we  can  either  find  x  y  in  terms  of  p  0, 
or  p  6  in  terms  of  x  y.  In  applying  them,  strict  atten- 
tion must  be  paid  to  the*  sign  of  «,  according  to  the  con- 
vention named  above. 

Note.  —  We  have  confined  the  discussion  of  this  case  to  the 
change  from  rectangular  axes,  as  this  alone  is  of  very  frequent 
occurrence  in  practice.  It  may  be  well,  however,  to  give  the 
formulas  of  the  general  case,  in  which  «  is  supposed  to  have  any 
value  whatever.  Assuming,  in  the  above  diagram,  the  angle  YOX 
to  be  oblique,  we  should  have  in  the  triangle  OMP,  (Trig.,  867,) 

p:  x  ::  sm  u  :  sm  {w  —  (0  +  a)}     .'.     x- 

•     fa    ,      N 
p  :  y  :  :  sm  u  :  sm  (0  +  «)  y= 


gn 
P  sin  (e  + 


These  formulas  evidently  become  those  obtained  above,  when  «  = 
90°. 


TRANSFORMATION  OF  CO-ORDINATES.  63 

Corollary. — If  the  axis  of  x  is  taken  as  the  initial  line, 
a  —  0  ;  and  we  have 

x  =  p  cos  0, 

y  =  p  sin  0 : 

a  set  of  formulae  in  very  extensive  use. 

58.  Case  Fourth:    To   combine  a  change   of   origin 
with  any  other  transformation. 

To  effect  this,  we  first  apply  the  formulae  of  Art.  55, 
and  thus  pass  to  the  new  origin  with  a  system  of  axes 
parallel  to  the  primitive.  That  is,  in  effect,  we  remove 
the  original  system  to  the  new  position  which  the  proposed 
origin  requires.  The  formulae  for  the  special  transforma- 
tion in  hand  are  then  applied,  and  the  whole  change  is 
accomplished. 

From  the  nature  of  the  formulae  in  Art.  55,  it  is  obvious 
that  the  present  case  is  solved  analytically  by  merely 
adding  to  the  expressions  for  x  and  ?/,  the  co-ordinates 
m  and  n  of  the  new  origin.  Thus,  in  general,  by  Art.  56, 

X  sin  (co  —  a)  4-  Y  sin  (to  —  dl) 
x  =  m  -f  /     / — ^—  — , 

sin  -f-coj 

.   X  sin  a  -f  Y  sin  8 
»"";  +  ;       -i5^r       ' 

To  pass  from  bilinears  to  polars  when  the  pole  is  a  dif- 
ferent point  from  the  origin,  we  have 

x  =  m  -f-  p  cos  (0  -f  a), 
y  =  n  -\-  p  sin  (6  -f-  a). 

59.  It   should  be   observed   that   in   applying   these 
various  formulae,  great  care  is  to  be  exercised  in  respect 
to  the  signs  of  the  constants  involved.     And  in  the  ex- 
amples which  follow,  where  given  points  are  to  be  trans- 
formed, the  same  care  should  be  taken  with  respect  to  all 

An.  Go.  9. 


64  ANALYTIC  GEOMETRY. 

the  known  co-ordinates.  It  is  recommended  that  the 
student  reduce  every  problem  to  drawing,  at  least  in  the 
earlier  stages  of  his  studies.  In  no  other  way  will  he 
readily  acquire  the  habit  of  bringing  every  analytic 
process  to  the  test  of  geometric  interpretation. 

EXAMPLES. 

1.  Given  the  point  (5,  6) :  what  are  its  co-ordinates  for  parallel 
axes  through  the  origin  (2,  3)  ? 

2.  Transform  (— 3,  0)  to  parallel  axes  through  (—4,  —  5);— 
to  parallel  axes  through  (5,  —3); — through  (—3,  5); — through 
(-3,0). 

3.  Given  in  rectangular  co-ordinates  the  points  (1,  1),  ( —  1,  1), 
(2,  —  1),  and  (—3,  — 3):  find  their  polar  co-ordinates,  the  origin 
being  the  pole,  and  the  axis  of  x  the  initial  line. 

4.  Transform  the  points  in  Ex.  3,  supposing  the  pole  to  be  at 
( —  4,  5),  and  the  initial  line  to  make  with  the  axis  of  #  an  angle 
„  =  _  30°. 

5.  Solve  Ex.  4,  on  the  supposition  that  a  =  45°. 

6.  Find  the  rectangular  co-ordinates  of  (P  =  3;  6  =  60°)  and 
(p  =  —  3 ;  0  =  —  60°),  the  origin  and  axis  of  x  coinciding  respect- 
ively with  the  pole  and  the  initial  line.     Find  the  same,  supposing 
the  origin  at  (—2,  —  1),  and  the  angle  a  =  30°. 

7.  The  co-ordinates  of  a  point  for  a  set  of  axes  in  which  u  = 
60°,  satisfy  the  equation  Sx  -f-  4?/  —  8  —  0:   what  will  the  equation 
become  when  transformed  to  w'  =  45°,  a  =  15°?     What,  when  in 
addition  the  origin  is  moved  to  ( —  3,  2)  ? 

8.  Transform  x2  -f  y1  —  r"1  to  parallel  axes  through  (—a,  —I). 

9.  It  is  evident  that  when  we  change  from  one  set  of  rectangular 
axes  to  another  having  the  same  origin,  x1  -f~  yl  must  be  equal  to 
xn  -j-  3/2,  since  both  express  the  square  of  the  distance  of  the  point 
from  the  origin.    Verify  this  by  squaring  the  expressions  for  x  and 
y  given  in  Art  56,  Cor.  3,  and  adding  the  results. 

10.  Transform  to  rectangular  co-ordinates  the  following  equations 
in  polar ;  origin  same  as  pole,  and  axis  of  x  as  initial  line : 

c*;     p2  =  c2  cos  2  (9. 


INTERPRETATION  IN  GENERAL.  65 

GENERAL    PRINCIPLES    OF   INTERPRETATION.  ' 

GO.  In  an  important  sense,  the  whole  science  of 
Analytic  Geometry  may  be  said  to  consist  in  knowing 
how  to  translate  algebraic  symbols  into  geometric  facts. 
Supposing  that  we  have  solved  any  geometric  problem 
by  analytic  methods,  our  result  must  be  some  algebraic 
expression.  Hence,  in  the  end  the  question  is,  How 
shall  we  interpret  that  expression  into  the  geometric 
property  which  we  are  seeking  ? 

The  principles  governing  such  an  interpretation  are  to 
be  mastered  in  their  fullness,  only  through  an  exhaustive 
study  of  the  whole  field  of  Analytic  Geometry.  But 
there  are  a  few  of  them,  which,  lying  at  the  root  of  all 
the  others,  are  of  universal  application,  and  must  be  de- 
termined at  the  outset.  In  the  following  articles,  we 
will  state  and  establish  them.  The  student  may  notice 
that  the  illustrations  and,  in  some  cases,  the  phraseology 
refer  to  Cartesian  co-ordinates ;  but  this  does  not  affect 
the  generality  of  the  principles,  since  we  can  convert  any 
geometric  expression  into  one  relating  to  the  Cartesian 
system  by  transformation  of  co-ordinates. 

61.  A  single  equation  betiveen  plane  co-ordinates  repre- 
sents a  plane  IOCHS. 

This  theorem  follows  directly  from  the  corollary  of 
Art.  27.  It  may  be  well,  however,  to  add  here  some 
illustrations  of  the  principle. 

It  is  plain  that  if  we  have  any  equation  between  two 
variables,  as 

ax*  -f  bx*y  -f  cxf  +  dtf  +/=  0, 

we  may  assign  to  x  any  value  we  please,  and  obtain  a 
corresponding  value  for  y.  The  unknown  quantities  in 
such  an  equation  are  therefore  not  determinate.  On  the 
contrary,  there  is  a  series  of  values,  infinite  in  number, 


66  ANALYTIC   GEOMETRY. 

any  of  which  will  satisfy  the  equation.  Taking  these 
values  as  denoting  the  co-ordinates  of  a  point,  the  equa- 
tion must  represent  an  infinite  number  of  points.  But, 
though  infinite  in  number,  these  points  can  not  be  taken 
at  random;  for  the  equation  can  not  be  satisfied  by 
values  arbitrary  for  y  as  well  as  x,  but  only  by  such 
values  of  y  as  its  own  conditions  require  in  answer  to 
the  assigned  values  of  x.  Hence,  the  infinite  series 
of  points  which  the  equation  represents,  conforms  in  all 
its  members  to  the  same  law  of  position :  a  law  ex- 
pressed in  the  uniform  relation  which  the  equation 
establishes  between  the  values  of  its  variables.  Such  a 
series  of  points  must  constitute  a  line. 

To  illustrate  by  a  diagram, 
we  may  suppose  that  in  a 
given  equation  between  two  va- 
riables, x  has  the  value  Om. 
Corresponding  to  this  there  will 
be,  let  us  say,  three  values  of  y, 
represented  by  mp,  mq,  mr.  We 
thus  determine  three  points,  p, 

q,  r.  Again,  supposing  x  =  Om' ,  we  determine  three 
other  points,  pf,  qf,  /.  And  again,  making  x  --  Om" , 
we  obtain  the  points  p",  q",  r".  We  may  continue  this 
process  as  long  as  we  please,  and  determine  any  number 
of  points,  by  assigning  successive  values  to  x.  By  taking 
these  sufficiently  near  each  other,  and  drawing  a  line 
through  the  points  thus  found,  we  may  determine  the 
figure  of  the  locus  which  the  equation  represents. 

Remark. — We  must  here  carefully  recall  the  proposition,  stated 
in  the  Introduction,  that  to  render  the  principle  of  this  article  uni- 
versally true  we  must  take  into  account  imaginary  loci,  loci  at 
infinity,  and  cases  where  a  locus  degenerates  into  disconnected  points 
or  a  single  point.  For  example,  the  equation 
x*  +  y*  +  1  =  0 


INTERPRETATION  IN  GENERAL.  67 

can  not  be  satisfied  by  any  real  values  of  x  and  y :  consequently,  in 
order  to  bring  it  within  the  terms  of  our  principle,  we  must  say  that 
it  denotes  an  imaginary  locus.     It  should  be  borne  in  mind  that 
this  amounts  to  saying  that  it  has  no  geometric  locus. 
Again,  the  equation 

Ox  +  Oy  +  c  =  0 

can  be  satisfied  by  none  but  infinite  values  of  x  and  ?/.  All  the 
points  on  its  locus  are  therefore  at  an  infinite  distance  from  the 
origin,  and  it  can  be  brought  within  the  terms  of  our  principle  only 
by  saying  that  it  denotes  a  locus  at  infinity.  This,  too,  is  only  another 
way  of  saying  that  it  has  no  geometric  locus. 
Again,  the  equation 

(x  -  a)  2  +  7/2  =  0 
obviously  can  not  be  satisfied  unless  we  have,  at  the  same  time, 

(x-  a)2=0  and  if  =  0; 
that  is,  it  admits  of  no  values  except 

x  =  a  and  y  =  0. 

Accordingly,  if  we  would  bring  it  within  our  principle,  we  must  say 
that  it  denotes  a  locus  which  has  degenerated  into  a  point  situated  on 
the  axis  of  x,  at  a  distance  a  from  the  origin.  We  shall  learn  here- 
after that  this  point  is  an  infinitely  small  circle,  having  (a,  0)  for 
its  center. 

62.  Any  two  simultaneous  equations  between  plane  co- 
ordinates represent  determinate  points  in  a  given  plane. 

For,  given  two  equations  between  two  variables,  we 
can  determine  the  values  of  x  and  y  by  elimination. 

Moreover,  the  points  which  such  a  pair  of  simultaneous 
equations  determines,  are  the  points  of  intersection  common 
to  the  two  lines  which  the  equations  respectively  repre- 
sent. For  it  is  obvious  that  the  values  of  x  and  y  found 
by  elimination,  must  satisfy  both  of  the  equations  ;  hence, 
the  points  which  these  values  represent  must  lie  on  both 
of  the  lines  represented  by  them :  that  is,  they  are  the 
points  common  to  those  lines. 


68  .ANALYTIC  GEOMETRY. 

Corollary  1. — Hence,  To  find  the  points  of  intersection 
of  tivo  lines  given  by  their  equations,  solve  the  equations 
for  x  and  y. 

Remark. — T he  geometric 
meaning  of  simultaneity  and 
elimination  may  be  made  clearer 
by  the  accompanying  diagram. 

Let  the  curve  A  be  represented        -    /Q£,    ^^ % 

by  the  equation 


and  the  curve  B  by  a  second  equation 

y  =  ?(*)*  (2): 

then,  supposing  A  to  intersect  B  in  the  points  pr  and  pff, 
and  in  these  points  only,  it  is  obvious  that  the  x  and  y 
of  equation  (1)  will  become  identical  with  the  x  and  y 
of  equation  (2)  when  we  substitute  in  both  equations  the 
co-ordinates  of  p'  and  p" ;  for  we  shall  then  have,  in  both 
equations, 

x  =  Om'  or  Om", 

y  =  m'p'  or  mrfp". 

And  it  is  equally  plain  that  the  x  and  y  of  the  two  equa- 
tions will  not  be  the  same,  but  different,  so  long  as  they 
represent  the  co-ordinates  of  any  other  point.  Thus,  if 
in  (2)  we  make  x  =  OM,  we  shall  have  y  =  MP :  values 
which,  it  is  manifest  from  the  figure,  the  x  and  y  of  (1) 
can  not  have. 

Since,  then,  the  variables  in  the  equations  to  different 
curves  will  in  general  have  different  values ;  and  since, 
even  in  curves  that  intersect,  the  variables  in  their  re- 
spective equations  will  become  identical  in  value  only  at 


# Equations  (1)  and  (2)  are  read  "y  =  any  function  of  x"  and  "y 
any  other  function  of  x." 


INTERPRETATION  IN  GENERAL.  69 

the  points  of  intersection :  we  learn  the  important  principle, 
that  to  suppose  two  geometric  equations  simultaneous  is 
to  suppose  that  their  loci  intersect.  In  short,  simultaneity 
means  intersection ;  and  elimination  determines  the  intersect- 
ing points. 

Corollary  2, —  Two  equations,  of  the  mth  and  nth  degree 
respectively,  represent  mn  points. 

For  (Alg.,  246),  elimination  between  them  involves 
the  solution  of  an  equation  of  the  mnth  degree ;  and  such 
an  equation  (Alg.,  396,  397)  will  have  mn  roots. 

Two  lines,  therefore,  of  the  mth  and  wth  order  respect- 
ively, intersect  in  mn  points.  Two  lines  of  the  first  order, 
for  example,  have  but  one  point  of  intersection ;  two  of 
the  second,  have  four ;  a  line  of  the  first  order  intersects 
one  of  the  second,  in  two  points ;  a  line  of  the  second 
order  cuts  one  of  the  third,  in  six;  and  so  on. 

It  should  be  observed  that  any  number  of  these  mn 
points  may  become  coincident :  a  fact  which  will  be  indi- 
cated, of  course,  by  the  existence  of  a  corresponding 
number  of  equal  roots  in  the  equation  obtained  by  elimi- 
nation. Or,  any  number  of  them  may  become  imaginary ; 
or,  in  certain  cases,  be  situated  at  infinity :  facts  respect- 
ively indicated  by  the  presence  of  imaginary  and  infinite 
roots. 

63.  An  equation  lacking  the  absolute  term  represents  a 
line  passing  through  the  origin. 

For  every  such  equation  will  be  satisfied  by  the  values 
x  =  0,  y  =  0 ;  and  these  (Art.  49,  Cor.  3)  are  the  co- 
ordinates of  the  origin. 

64.  Transformation  of  co-ordinates  does  not  alter  the 
degree  of  an  equation,  nor  affect  the  form  of  the  locus^which 
it  represents. 


70  ANALYTIC  GEOMETRY. 

For,  supposing  the  equation  to  be  originally  of  the 
?ith  degree,  the  term  which  tests  that  degree  may  be 
written  Mxrys^  in  which  r  -j-  s  =  n.  Now  the  most  gen- 
eral case  of  transformation  (compare  Arts.  56,  58)  will 
require  us  to  substitute  for  this  an  expression  of  the 
form 

M(aX+  IY+  ey  (a'X+  b'Y+  <T  : 

which  when  expanded  will  certainly  contain  terms  of  the 
form  MfXpYq,  where  p  -{-  q  =  r  -f  s  =  n,  but  can  contain 
none  in  which  the  sum  of  the  exponents  of  -X"  and  Y  is 
greater  than  n.  Hence  the  degree  of  the  equation,  and 
therefore  the  order  of  its  locus,  will  remain  unchanged 
through  any  number  of  transformations. 

Nor  will  transformation  affect  the  form  of  the  locus  at 
all.  For,  obviously,  the  figure  of  a  curve  does  not  depend 
upon  limits  to  which  we  arbitrarily  refer  its  points. 

EXAMPLES. 

1.  What  point  is  represented  by  the  equations  3x  +  5y  =  13 
and  4#  — y  =  2? 

2.  Given  the  two  curves  x2  -f-  y2  =  5  and  xy  =  2,  in  how  many 
points  will  they  intersect?     Find  the  points  of  intersection. 

3.  Find  the  points  in  which  x  —  y  =  1  intersects  x1  -f-  3/2  =  25. 

4.  Of  what  order  is  the  curve  y2  =  4px?     Show,  by  actual  trans- 
formation, that  it  continues  of  the  same  order  when  passed  from  its 
original  rectangular  axes  to  oblique  ones  through  «,  iVpci'-    the 
new  axis  of  x  being  parallel  to  the  old,  and  the  inclination  of  the 
new  axes  being  the  angle  whose  tan2  =p  :  a. 

5.  Decide  whether  the  following  curves  pass  through  the  origin : 

y  =  mx  -f  I ;  x1  —  ?/2  —  0 ;   3?  —  if  =  1  ;  y1  =  4px  ; 
3.r3  -  ~).v>/  4-  lx*  -  Sy  =  0. 


SPECIAL  INTERPRETATION.  71 

SPECIAL  INTERPRETATION  OF  PARTICULAR  EQUATIONS. 

65.  The   foregoing  principles  illustrate  the  doctrine 
that  there  is  a  general  connection  between  an  equation 
and  the  locus  of  a  point.     But  every  curve  *  has  its  own 
particular  equation,  and  we  may  appropriately  close  our 
discussion  of  the  Theory  of  Points  by  explaining  briefly 
how  to  discover  the  form  and  situation  of  a  curve  from 
its  equation. 

66.  The  Special  Interpretation  of  an  Equation 

consists  in  tracing,  by  means  of  determined  points,  the 
curve  which  it  represents. 

67.  In  order  to  trace  any  curve  from  its  equation,  we 
solve  the  equation  for  either  of  its  variables,  say  for  y. 
We  then  assign  to  x  various  values  at  pleasure,  and  com- 
pute the  corresponding  values  of  y.     Then,  drawing  the 
axes,  we  lay  down  the  points  corresponding  to  the  co- 
ordinates thus   found.     A   curve   traced   through   these 
points  will   approximately   represent   the   locus    of   the 
equation.     Could  we  take  the  points  infinitely  near  each 
other,  we  should  obtain  the  exact  curve. 

68.  Attention  to  certain  characteristics  of  the  given 
equation  and  of  the  values  of  the  variable  for  which  it  is 
solved,  will  enable  us  to  decide  certain  questions  con- 
cerning the  peculiar  form  of  the   corresponding    curve. 
These    algebraic    characteristics,    and    their    geometric 
meaning,  AVC  will  now  specify. 

69.  If  the  given  equation  is  of  a  degree  higher  than 
the  first,  for  every  value  assigned  to  x  there  will  arise 


*It  is  customary  to  call  any  plane  locus  a  curve,  even  though  this  in- 
volves the  apparent  harshness  of  saying  that  the  right  line  or  an  isolated 
point  is  a  curve. 


72  ANALYTIC   GEOMETRY. 

two  or  more  values  of  y.    The  several  points  correspond- 
ing to  the  common  abscissa  are 
said  to  lie  on  different  PORTIONS 
of  the  curve.   Thus,  in  the  figure, 
the  points  p,  p',  p"  lie  on  one 
portion  of  the  curve  represented; 
the  points  q,  qr,  q"  on  another; 
and  the  points  r,  r',  r"  on  a  third. 
The  LIMITS  of  a  portion  —  that 
is,  the  points  where   it  merges  into  another  portion  — 
are  the  points  whose  abscissas  cause  two  values  of  the 
ordinate  to  become  equal. 

Corollary. — Hence,  To  test  whether  a  curve  consists  of 
several  portions,  note  whether  its  equation  is  of  a  degree 
higher  than  the  first.  To  find  the  limits  of  the  portions, 
observe  what  values  of  x  give  rise  to  equal  roots  for  y. 

TO.  If  all  the  values  assigned  to  x  within  the  limits 
separating  two  portions  of  a  curve,  make  the  y'a  of  the 
two  portions  numerically  equal  but  of  opposite  sign,  the 
corresponding  points  of  these  portions  will  be  equally 
distant  from  the  axis  of  x.  Similar  conditions  with 
respect  to  the  axis  of  y  will  determine  points  equally 
distant  from  that  axis. 

Two  portions  of  a  curve,  whose  points  are  thus  situated 
with  reference  to  either  axis,  are  said  to  be  SYMMETRICAL 
to  that  axis.  A  curve  is  symmetrical,  when  all  its  por- 
tions taken  two  and  two  are  symmetrical. 

Corollary. — Hence,  To  test  for  symmetry,  note  whether  the 
values  of  either  variable,  corresponding  to  all  values  of  the 
other  between  the  limits  of  two  portions  of  a  curve,  appear 
in  pairs,  numerically  equal  with  contrary 


71.  If  any  value  assigned  to  x  gives  rise  to  imaginary 


SPECIAL  INTERPRETATION.  73 

values  for  y,  the  corresponding  point  or  points  will  be 
imaginary.  That  is,  the  curve  is  interrupted  at  such 
points.  And  if,  between  any  two  values  of  either  vari- 
able, the  corresponding  values  of  the  other  are  all 
imaginary,  the  curve  does  not  exist  between  the  corre- 
sponding limits.  Thus,  in  the  curve 


by  solving  for  y  we  obtain 


y=        -Vx2—a2: 
a 

so  that  y  is  real  for  every  value  of  x  which  lies  beyond 
the  limits  x  =  a  and  x  =  —a,  but  is  imaginary  for  every 
value  of  x  lying  between  them  ;  and  the  curve  is  interrupted 
in  the  latter  region. 

When  the  extent  of  a  curve  is  nowhere  interrupted, 
and  it  suffers  no  abrupt  changes  in  curvature,  *  it  is  said 
to  be  CONTINUOUS.  A  curve  may  be  either  continuous 
throughout  or  composed  of  continuous  parts. 

Corollary,  —  To  test  for  continuity  in  extent,  note  whether 
the  equation  to  a  curve  gives  rise  to  limiting  values  of  either 
variable,  beyond  or  between  which  the  values  of  the  other 
are  imaginary. 

Remark,  —  To  test  for  continuity  in  curvature,  we 
employ  the  Differential  Calculus. 


The  continuous  parts  of  a  curve  are  called  its 
BRANCHES.  A  branch  should  be  distinguished  from  a 
portion  of  a  curve  :  a  branch  may  consist  of  several 
portions;  or  a  portion,  of  several  branches. 


*  Cumaturc  i.  e.  the  rate  at  which  a  curve  deviates  from  a  riylit  line. 


74  ANALYTIC  GEOMETRY. 

A  branch  of  a  curve  may  degenerate  into  isolated 
points,  or  a  single  point:  such  points  are  called  CONJU- 
GATE POINTS. 

Corollary. — The  number  and  extent  of  the  branches 
belonging  to  a  curve  may  often  be  determined  by  exam- 
ining the  limits  beyond  or  between  which  its  equation  gives 
rise  to  imaginary  values  of  the  variables.  Thus,  if 


y  will  be  real  for  all  values  of  x  lying  between  the  limits 
x  =  —  a  and  x  =  a,  but  imaginary  for  all  lying  beyond. 
The  curve  therefore  consists  of  a  single  branch,  surround- 
ing a  portion  of  the  axis  of  x  wrhose  length  =  2a.  If 

_  _^_  b        2 

a 

y  is  real  for  all  values  of  x  lying  beyond  the  limits 
x  =  —  a  and  x  =  a,  but  imaginary  for  all  lying  between. 
Hence,  the  curve  consists  of  two  branches,  separated  by 
a  portion  of  the  axis  of  x  whose  length  =  2a.  If 

y  =  2  Vpx, 

y  is  imaginary  for  all  negative  values  of  x,  but  real  for 
all  positive  values.  Hence,  the  curve  consists  of  a  single 
infinite  branch,  extending  from  the  origin  toward  the 
right. 

Remark. — Conjugate  points  belong  to  a  class,  known 
as  SINGULAR  POINTS,  whose  existence  can  not  in  general 
be  tested  without  the  aid  of  the  Differential  Calculus. 
If,  however,  a  given  equation  is  obviously  satisfied  by 
none  but  isolated  values  between  certain  limits,  its  locus 
between  those  limits  will  consist  of  conjugate  points. 


SPECIAL  INTERPRETATION.  75 

73.  We  add  a  few  examples,  merely  premising  that 
it  is  often  convenient  first  to  find  the  situation  of  the 
axes  to  which  the  given  equation  is  referred.  This  is 
done  by  making  the  x  and  y  of  the  equation  suc- 
cessively equal  to  zero :  the  resulting  values  of  y  and  x 
(Art.  62,  Cor.  1)  are  the  intercepts  made  by  the  curve 
on  the  axis  of  y  and  of  x  respectively. 

We  have  given,  for  the  sake  of  widening  a  little  the 
student's  view  of  the  subject,  a  few  equations  to  Higher 
Plane  Curves,  both  Algebraic  and  Transcendental.  These 
curves,  of  course,  are  beyond  the  province  of  the  present 
work;  and  the  reader  who  desires  full  information  in 
regard  to  them  is  referred  to  SALMON'S  Higher  Plane 
Curves  or  to  the  writings  of  PLUCKER,  PONCELET,  and 
CHASLES. 

It  will  be  most  convenient,  in  tracing  the  curves  of  the 
following  examples,  to  use  paper  ruled  in  small  squares, 
whose  constant  side  may  be  taken  for  the  linear  unit. 
Let  the  limits  of  the  imaginary  values,  if  such  exist  in 
any  equation,  be  first  found :  then,  within  the  sphere  of 
real  values,  let  the  abscissas  be  taken  near  enough 
together  to  determine  the  figure  of  the  curve. 

The  axes  are  rectangular:  as  we  shall  always  suppose 
them  in  examples,  unless  the  contrary  is  indicated. 

EXAMPLES. 

1.  Represent  the  curve  denoted  by  y  —  2x  +  3. 
Making,  successively,  x  =  0  and  y  =  0,  we  obtain 

y  =  3  and  x  =  -     . 


The  curve  therefore  cuts  the  axis  of  y  at  a  distance  3  above  the  origin, 

g 
and  the  axis  of  x  at  a  distance  -  to  the  1 

and  lay  down  the  corresponding  points. 


and  the  axis  of  x  at  a  distance  —  to  the  left  of  the  origin.     Draw  the  axes 


76  ANALYTIC   GEOMETRY. 

The  equation  being  of  the  first  degree,  the  curve  consists  of  but  one 
portion,  y  is  obviously  real  for  all  real  values  of  x  :  the  curve  is  there- 
fore of  infinite  extent.  Making  x  successively  —  3,  —  2,  —  1,  1,  2,  3,  the 
corresponding  values  of  y  are  —  3,  —  1,  1,  5,  7,  9.  Laying  down  the 
points 

(-  3,  -  3),  (-  2,  -  1),  (-  1,  1),  (1,  5),  (2,  7),  (3,  9), 


we  find  that  they  all  come  upon  the  right  line  drawn  through  (0,  3)  and 
*s  therefore  the  curve  represented  by  the  given  equation. 


if1     /  ' 


2.  Interpret  ~  +  |-  =  1. 
Making  x  =  0,  we  obtain 


or  the  curve  cuts  the  axis  of  y  in  two  points:  one  at  the  distance  2  above 
the  origin,  and  the  other  at  the  same  distance  below  it. 

Making  y  =  0,  we  obtain 


whence  the  curve  cuts  the  axis  of  x  in  two  points  equally  distant  from 
the  origin,  and  on  opposite  sides  of  it. 

Since  the  equation  is  of  the  second  degree,  the  curve  consists  of  two 
portions;  and  as  the  values  of  y  coincide  and  =  0  when  x.  =  ±  3,  these 
portions  are  separated  by  the  axis  of  x.  Solving  for  y,  we  find 

o 
y  =  ±  3  1/9  -  x2: 

hence,  y  will  become  imaginary  when  x  >  3  or  x  <  —  3.  The  curve 
therefore  has  no  point  beyond  its  intersections  with  the  axis  of  x.  But 
for  every  value  of  x  between  the  limits  —3  and  3,  y  is  real;  that  is,  the 
curve  consists  of  a  single  continuous  branch. 

Making,  now,  x  successively  equal  to  —  3,  —  2.5,  —  2,  —  1,  0,  1,  2,  2.5,  3, 
the  corresponding  values  of  y  are  0,  ±  1.2, 
±  1.5,  ±  1.9,  ±  2,  ±  1.9,  ±  1.5,  ±  1.2,  0.  From 
these  values,  we  see  that  the  curve  is  sym- 
metrical to  both  axes ;  and,  laying  down  the 
sixteen  points  thus  found,  we  determine  the 
figure  of  the  curve  as  annexed.  It  is  an 
ellipse. 

3.  Interpret  the  equation  y  —  mx. 

x2-       y2 

4.  Interpret  ^ ~  =  1.     This  curve  is  an  hyperbola. 

5.  Interpret  y*  =  4#.     This  curve  is  a  parabola. 


THE  EIGHT  LINE.  77 

6.  Interpret  y  =  x3  and  y2  =  xz.     The  first  of  these  curves   is 
a  cubic  parabola;  and  the  second,  a  semi-cubic  parabola. 

7.  Interpret  xz  —  (a  —  x)  y2-  =  0.     This  is  known  as  the  Cissoid 
of  Diodes. 

8.  Interpret  x*if  =  (a2  —  y2}  (b  +  y)2.      This   is   the   Conchoid 
of  Nicomedes. 

9.  Interpret  x  =  versin  -1  y  *  —  1/2  ry  —  y'L.     This  is  the  Com- 
mon Cycloid. 

10.  Interpret  y  —  sin  ar,  the  Ckrve  o/  /Smes ;  y  =  cos  #,  the 
o/"  Cosines ;  and  y  —  log  x,  the  Logarithmic  Curve. 


SECTION  II.  —  THE  RIGHT  LINE. 

THE    RIGHT   LINE    UNDER   GENERAL    CONDITIONS. 

74.  In  discussing  the  mode  of  representing  the  Right 
Line  by  analytic  symbols,  we  shall  in  the  first  place  have 
to  determine  the  various  forms  of  the  equation  which  rep- 
resents any  right  line;  that  is,  our  problem  will  be  to 
represent  the  Right  Line  under  general  conditions  only. 
This  accomplished,  we  shall  then  pass  to  the  more  partic- 
ular forms  of  the  equation  —  those  which  represent  the 
Right   Line    under   such   special   conditions    as   passing 
through  two  given  points,  passing  in  a  given  direction 
through  one  given  point,  etc. 

75.  In  showing  that  a  certain  curve  is  represented  by 
a  certain  equation,  we  may  proceed  in  either  of  two  ways. 


*  This  is  the  notation  for  an  inverse  triyonomct.-ic  function,  and  is  in  this 
case  read  "  the  arc  whose  versed-sine  is  y."  Similar  expressions  occur  in 
terms  of  the  other  trigonometric  functions  :  as,  x  —  sin—1  y;x  =  tan~a  y  ; 
etc.,  read  "  x  =  the  arc  whose  sine  is  y,"  "  x=  the  arc  whose  tangent  is  y," 
and  so  on. 


78  ANALYTIC   GEOMETRY. 

First,  we  may  begin  by  assuming  some  fundamental 
property  of  the  curve,  define  the  curve  by  means  of  it, 
and,  with  the  help  of  a  diagram  which  brings  it  into  rela- 
tion with  elementary  geometric  theorems,  embody  it 
in  an  equation  between  the  co-ordinates  of  any  point 
on  the  curve  —  an  equation  from  which  all  other  prop- 
erties may  be  deduced  by  suitable  transformations. 
Or,  secondly,  we  may  begin  without  any  geometric 
assumptions  except  those  on  which  the  convention  of 
co-ordinates  is  founded;  may  take  an  equation  of  any 
degree,  in  its  most  general  form;  and,  by  the  purely 
analytic  processes  of  algebraic,  trigonometric,  or  co- 
ordinate transformation,  reduce  the  equation  to  such 
simpler  forms  as  will  show  us  the  species,  figure,  and 
properties  of  the  corresponding  curve.  The  latter 
method  is  the  purely  analytic  one ;  the  former  mingles 
the  processes  of  geometry  and  analysis. 

TO.  In  the  present  Book,  both  of  these  methods  will 
be  applied  in  succession.  It  will  be  natural  to  set  out 
from  the  geometric  point  of  view:  for  in  this  way  we 
shall  secure  simplicity  and  clearness,  by  constantly 
bringing  the  analytic  formulae  and  operations  to  the 
test  of  interpretation  by  a  diagram.  After  the  char- 
acteristic forms  of  the  equation  to  any  locus  have  been 
obtained  by  the  aid  of  geometry,  and  the  beginner 
has  become  familiar  with  their  geometric  meaning, 
he  may  safely  ascend  to  the  higher  analytic  stand- 
point, and  will  be  able  to  descend  from  it  with  some 
real  appreciation  of  the  scientific  beauty  which  it  brings 
to  light. 

We  proceed  to  apply  to  the  Right  Line  the  two 
methods  mentioned,  and  shall  follow  the  order  which 
has  just  been  indicated. 


EQUATION  TO  EIGHT  LINE.  79 

I.    GEOMETRIC    POINT    OF    VIEW  :  —  THE    EQUATION    TO    THE 
RIGHT   LINE    IS    ALWAYS    OF    THE    FIRST    DEGREE. 

TT.  There  are  three  principal  forms  of  the  equation 
to  the  Right  Line,  arising  out  of  the  three  sets  of  data 
by  which  the  position  of  the  line  is  supposed  to  be  deter- 
mined. Each  of  these  will  prove  to  be  of  the  first 
degree. 

78.  Equation  to  the  Right  Line  in  terms  of  its 
angle  wills  the  axis  of  .r  and  its  intercept  on  the 
axis  of  ?y. — It  is  obvious,  on  inspecting  the  diagram,  that 
the  position  of  the  line  DT  is  given 
when  the   angle  DTX*  and  the 
intercept  OD  are  given. 

Let  x  and  y  denote  the  co-ordi- 
nates OM,  MP  of  any  point  P  on 
the  line.  Let  the  angle  DTX=  a, 
and  let  OD  =  6.  Then,  drawing 
MR  parallel  to  DT,  we  shall  have 
(Trig,  867) 

OR  :  OM  : :  sin  OME  :  sin  OEM. 
That  is,         b  —  y  :  x  : :  sin  a,  :  sin  (« —  to). 

sin  a.  .    7 

•'•  »  =  sr£r=-5j*  +  J; 

or,  putting  m  =  -. — sm  a , 

sm  (co  —  a) 

y  =  mx  -f  b: 
which  is  the  equation  in  the  terms  required. 


~-::~  The  student  will  not  fail  to  notice  that  the  angle  which  a  lino  makes 
with  the  axis  of  x  is  always  measured,  positively,  from  that  axis  toward 
the  left;  an  angle  measured  from  the  axis  toward  the  right,  is  negative. 
This  principle  holds  true  of  the  angle  between  any  two  lines. 
An.  Ge.  10. 


80  ANALYTIC  GEOMETRY. 

Corollary  1, — In  the  equation  just  obtained,  the  axes 
are  supposed  to  have  any  inclination  whatever.  If  the 
axes  are  rectangular,  co  =  90°  and  m  =  tan  a.  Hence,  in 

y  =  mx  -\-  5, 

when  referred  to  rectangular  axes,  m  denotes  the  tangent 
of  the  angle  which  the  line  makes  with  the  axis  of  x; 
but  when  the  equation  is  referred  to  oblique  axes,  m  de- 
notes the  ratio  of  the  sines  of  the  angles  which  the  line 
makes  with  the  two  axes  respectively. 

Remark. — In  interpreting  an  equation  of  the  form 
y  =  mx  -\-  6,  and  tracing  the  line  corresponding  to  the 
values  m  and  b  have  in  it,  account  must  be  taken  of  the 
signs  of  those  constants. >:<  The  constant  m  will  be  posi- 
tive or  negative  according  as  the  angle  a  is  less  or  greater 

than  co.     For  m  =  ~  —  :  which  is  positive  or  neg- 

sin  (co  —  «) 

ative  upon  the  condition  named,  according  to  Trig.,  829 ; 
since  a  is  supposed  not  to  exceed  180°.  The  constant  b 
(Art.  49)  will  be  positive  or  negative  according  as  the 
intercept  on  the  axis  of  j  falls  above  or  below  the,  origin. 
Thus,  in  the  case  of  the  line  in  the  diagram,  m  is 
negative,  and  b  positive. 

If  the  axes  are  rectangular,  the  sign  of  m  will  be  -f-  or 
-  according  as  a.  is  acute  or  obtuse.     For,  in  that  case, 
m  =  tan  a ;  and  the  variation  of  sign  is  determined  by 
Trig.,  825. 

Corollary  2. — We  can  thus  determine  the  position  of  a 
right  line  with  respect  to  the  angles  about  the  axes,  by 
merely  inspecting  the  signs  of  its  equation. 


*  The  quantities  m  and  b  are  constants,  since  they  can  have  but  one 
value  for  any  particular  right  line.  But  the  equation  is  true  for  any  right 
line,  because  we  can  assign  to  m  and  6  any  values  we  please.  They  are 
hence  called  arbitrary  constants. 


EQUATION  TO  EIGHT  LINE.  81 

If  m  is  negative,  and  b  positive,  'the  line  crosses  the 
axis  of  y  above  the  origin,  and  makes  with  the  axis  of  x 
an  angle  greater  than  CD  :  it  therefore  crosses  the  latter 
at  some  point  to  the  right  of  the  origin,  and  so  lies  across 
the  first  angle. 

If  m  and  b  are  both  positive,  the  line  lies  across  the 
second  angle. 

If  m  and  b  are  both  negative,  the  line  lies  across  the 
third  angle. 

If  m  is  positive,  and  b  negative,  the  line  lies  across  the 
fourth  angle. 

[The  student  may  draw  a  diagram  and  verify  the  last 
three  statements.] 

Corollary  3. — If  m  =  0,  we  shall  have  sin  a  =  0 
.  • .  a  =  0,  and  the  line  will  be  parallel  to  the  axis 
of  x. 

Corollary  4. — If  m  =  oo,  sin  (to  —  a)  =  0  .  * .  a  =  a», 
and  the  line  will  be  parallel  to  the  axis  of  y. 

If  m  =  oo  when  the  axes  are  rectangular,  we  shall 
have  tan  a.  =  oo  .  • .  a  =  90°,  and  the  line  will  be  per- 
pendicular to  the  axis  of  x :  which  is  essentially  the  same 
result  as  before. 

Corollary  5, — If  b  =  0,  the  line  must  pass  through  the 
origin.  But,  in  that  case,  the  equation  becomes 

y  =  mx: 

which  is  therefore  the  equation  to  a  right  line  passing 
through  the  origin . 

79.  Equation  to  the  Right  Line  in  terms  of  its 
intercepts  on  the  two  axes. — The  diagram  shows  that 


82  ANALYTIC  GEOMETRY. 

the  position  of  DT-  is  given 
when  OT  and  OD  are  given. 
Let  OT=tf,and  OD=b;  rep- 
resent by  x  and  y,  as  before, 
the  co-ordinates  OM,  MP 
of  any  point  P  on  the  line. 
By  similar  triangles,  we  have 

OT:  OD  :  :  MT  :  MP  ;  that  is,  a  :  b  :  :  a  —  x  :  y. 


which  is  the  symmetrical  form  of  the  required  equation. 

Remark  1.  —  When  interpreting  an  equation  of  this 
form,  the  signs  of  the  arbitrary  constants  a  and  b  must 
be  observed.  By  doing  this,  we  can  fix  the  position 
of  the  line  with  respect  to  the  four  angles,  as  in  the 
preceding  article. 

When  a  and  b  are  both  positive,  the  line  lies  in  the 
first  angle,  as  in  the  diagram. 

When  a  is  negative,  and  b  positive,  the  line  lies  in  the 
second  angle. 

When  a  and  b  are  both  negative,  the  line  lies  in  the 
third  angle. 

When  a  is  positive,  and  b  negative,  the  line  lies  in  the 
fourth  angle. 

Remark  2.  —  This  form  of  the  equation  to  the  Right 
Line  is  much  used  on  account  of  its  symmetry.  It  also 
deserves  to  be  noticed  on  account  of  its  resemblance  to 
the  analogous  equations  to  the  Conies,  which  we  shall  de- 
velop in  due  time.  It  is  applicable,  as  is  manifest  from 
the  investigation,  to  rectangular  and  oblique  axes  alike. 


EQUATION  TO  RIGHT  LINE.  83 

SO.  Equation  to  the  Right  Line  in  terms  of  its 
perpendicular  from  the  origin  and  the  angle 
made  by  the  perpendicular  with  the  axis  of  x.  — 

By   examining    the    diagram,   it 

becomes    evident    that    the    po- 

sition and  direction   of  DT  are 

given,  if  the  length  of  OR  per- 

pendicular to  Dl\  and  the  angle 

TOR  which  it  makes  with  OX, 

are  given.     Let  OR=  p;  and  let       /o 

the  angle  TOR  =  a,  whence  the 

angle  DOR  =  (o  —  «.     Then,  a  and  b  representing  the 

intercepts  OT  and  OD  as  before,  we  have  (Trig.,  859) 


a  =         -  •   b  = 


cos  a  cos  (at  —  a) 

Substituting  these  values  of  a  and  b  in  the  equation  of 
Art.  79,  we  obtain 

x  cos  ay  cos  (o)  —  a)  _  -, 

P  P 

Clearing  of  fractions, 

x  cos  a  -f-  y  cos  (at  —  a)  =  p: 
which  is  the  equation  sought. 

Remark.  —  The  co-efficients  of  x  and  y  in  the  foregoing 
equation  are  called  the  direction-cosines  of  the  line  which 
the  equation  represents.  In  using  this  form  of  the 
equation,  it  is  most  convenient  to  suppose  that  the 
angle  a  may  have  any  value  from  0  to  360°,  and  that 
the  perpendicular  p  is  always  positive  —  that  is,  (Art.  50, 
Note,)  measured  from  0  in  such  a  direction  as  to  bound 
the  angle.  This  convention  can  not  be  too  carefully 
remembered. 


84  ANALYTIC  GEOMETRY. 

Corollary.  —  If  the  axes  are  rectangular,  we  shall  have 
(Trig.,  841) 

x  cos  a  -f-  y  sin  a  =  p  : 

a  form  of  the  equation  having  the  greatest  importance, 
on  account  of  its  relations  to  the  Abridged  Notation. 

81.  The  three  forms  of  the   equation   to   the  Right 
Line  are  therefore  as  follows: 

y  =  mx  -f  b  (1), 


x  cos  a  -f-  y  cos  (to  —  a)  =  p  (3). 

They  are  all  of  the  first  degree.  Either  of  them  may 
be  derived  from  any  other,  by  merely  substituting  for 
the  constants  in  the  latter  their  values  in  terms  of  those 
involved  in  the  form  sought.  For  (see  diagram,  Art.  80) 
we  have 


b  P       A 

=  --  ;  a  •=  —    —  ;  o  = 


P 


— 

cos  a  cos  (<o  —  a) 

In  the  case  of  rectangular  axes,  we  shall  have  b  ===  -**-—  . 

sin  a 

82.  Polar  Equation  to  tlie  Right  Line.  —  Let  p 

and  6  be  the  co-ordinates  OP,  XOP  of  any  point  P 
on  the  line  PT.  Let  OR,  the  perpen- 
dicular from  the  pole  to  the  line,  —p; 
and  let  XOR,  the  angle  which  the  per- 
pendicular makes  with  the  initial  line, 
-  «.  Then  (Trig.,  858)  we  have 

OP  cos  ROP=  OR 

that  is, 

p  cos  (6  —  a)  —  p  : 

which  is  the  equation  required. 


POLAR  EQUATION  TO  RIGHT  LINE.  85 

Corollary. — If  the  initial  line  were  perpendicular  to  the 
right  line,  we  should  have  a  --  0,  and  the  equation 
would  become 

p  cos  6  =  p  : 

the  equation  to  a  right  line  perpendicular  to  the  initial  line. 

Remark, — In  applying  the  equation  of  the  present 
article,  it  will  be  convenient  to  regard  the  sign  of  the 
angle  a.  This  will  be  +  if  the  perpendicular  OR  falls 
above  the  initial  line;  but  — ,  if  it  falls  below. 

83.  To    trace    a    Right    Line. — The    most    direct 
method  of  solving  this  problem,  consists  in  finding  the 
intercepts  made  by  the  line  on  the  axes,  and  laying  them 
off  according  to  any  chosen  scale  of  equal  parts. 

Hence,  to  trace  a  right  line  given  by  its  equation : 
Make  the  j  and  x  of  the  equation  successively  —  0  :  the 
resulting  values  of  x  and  j  mil  be  the  intercepts  on  the 
axis  of  x  and  of  j  respectively.  Lay  off  on  the  axes 
these  intercepts,  and  the  line  drawn  through  their  extremi- 
ties ivill  be  the  line  required. 

Remark. — If  theUine  is  given  by  its  polar  equation, 
we  find  its  intercept  on  the  initial  line  by  making  0  =  0. 
When  this  and  the  perpendicular  from  the  pole  are  laid 
off,  we  draw  the  line  through  their  extremities. 

Note. — This  method  fails  when  the  line  passes  through  the  origin 
or  the  pole,  or  is  parallel  to  either  axis.  In  the  former  case,  in  the 
Cartesian  equation,  make  x  =  1,  construct  the  corresponding  ordi- 
nate,  and  join  its  extremity  to  the  origin ;  in  the  polar  system,  lay 
off  the  constant  vectorial  angle  of  the  line.  A  parallel  to  either 
axis  must  be  drawn  as  such,  at  the  distance  its  equation  requires. 

84.  We  add  a  few  miscellaneous   exercises    on  the 
foregoing  articles. 


86  ANALYTIC  GEOMETRY. 

EXAMPLES. 

1.  Across  which  of  the  four  angles  does  the  line  y  —  Bx  -\-  5 
lie?  —  the  line  y  —  —  6z  +  2?  —  the  line  y  —  —  2x  —  4?  —  the 
line  y  =  x  —  \1 

2.  What  is  the  situation  and  direction  of  the  line  y  =  x  f 

3.  Axes  being  oblique,  what  angle  does  the  line  y  =  x  make 
with  the  axis  of  x  ?  —  the  line  y  =  -pr  -f  2  ? 

4.  What  is  the  direction  of  the  line  y  =  4  ? 

5.  What  is  the  direction  of  the  line  y  —  ^  -f  6  ?    What  are  the 
intercepts  made  by  the  preceding  lines  on  the  axis  of  yf 

6.  Trace  the  line  y  =  5x  -f-  3  ;  —  the  line  y  —  x  =  0. 

7.  Trace  the  line  |  +  |  =  1  ;  —  the  line  |  -  |  -  1. 

8.  Trace  the  line  i  x  1/3  +1  y  =  I. 

9.  What  is  the  value  of  the  angle  a  in  the  line  of  the  previous 
example?  —  of  the  angle  «?  —  of    the    perpendicular  pi     [Here 

cos  a  =  -  1/3";  cos  («  —  a)  ==  -  .  ] 

10.  x  +  jr  y  =  3  is  in  the  form  x  cos  a  -f-  y  cos  («  —  a)  =  jtK 

2 

determine  the  values  of  p,  «,  and  w,  and  trace  the  line. 

11.  -tfl/^+03/^5  being  referred  to  rectangular  axes,  find 
2  & 

the  values  of  a  and  jo,  and  lay  off  the  line  by  means  of  them. 

12.  In  which  of  the  angles  lie  the  lines  -  +  '-=!,-      o  —  l> 

A    '       O  u          4 


13.  In  which  angle  does  the  line  5  x  i/3  —  -y—  21ie? 

14.  Trace  the  lines  />  cos  (0  —  45°)  =  8  and  p  sin  0  =  —  6.    What 
is  the  value  of  a  in  the  second  line,  and  on  which  side  of  the  initial 
line  is  it  measured  ? 

15.  Find  the  intercepts  of  the  line  5x  +  ly  —  9  =  0,  and  trace 
the  line. 


EIGHT  LINE  IN  STRICT  ANALYSIS.  87 

II.    ANALYTIC  POINT  OF  VIEW:  —  EVERY  EQUATION  OF  THE 
FIRST  DEGREE  REPRESENTS  A  RIGHT  LINE. 

85.  The  most  general  form  of  the  equation  of  the 
first  degree  in  two  variables,  is 

^  +  %  +  <7=o, 

in  which  A,  B  and  C  are  arbitrary  constants,  and  may 
have  any  value  whatever.  We  propose  to  prove  that 
this  equation,  no  matter  what  the  values  of  A,  £  and  C, 
always  represents  a  right  line. 

Solving  the  equation  for  y,  we  obtain 
A         C 

y-  ~SX~B' 

That  is,  y  is  always  equal  to  x  multiplied  by  an  arbitrary 
constant,  plus  an  arbitrary  constant.  In  other  words, 
the  equation  is  always  reducible  to  the  form 


and  therefore  (Art.  78)  always  represents  a  right  line. 

86.  A  second  proof  of  the  same  proposition,  by  means 
of  the  trigonometric  function  which  the  equation  implies, 
is  as  follows  : 

Write  the  equation  in  the  form  (to  which  we  have  just 
shown  that  it  is  always  reducible) 

y  —  mx  -f  b  : 

in  which  m  and  b  are  merely  abbreviations  for  the  arbitrary 

A  C 

constants  —  -^  and  —  -^  .     Now  the  equation,  being  true 

for  every  point  of  its  locus,  must  be  true  for  any  three 
points  x'y',  x"y",  x'"y"f,     Hence, 

y'    =mxf    +6  (1), 

y"  =  ™x"  +  ft  (2), 

y'"  =  mx"t-}-'b  (3). 

An.  Ge.  11. 


88 


ANALYTIC  GEOMETRY. 


Supposing,  then,  that  the  abscissas  are  taken  in  the 
order  of  magnitude,  the  equation  y  =  mx  -f  b  shows  that 
the  ordinates  will  also  be  in  the  order  of  magnitude ;  that 
is,  if  we  take  x"  greater  than  a/,  and  xf"  greater  than  x" , 
we  shall  either  have  y"  greater  than  y',  and  y"1  greater 
than  yh ',  or  else  y"  less  than  y',  and  y'"  less  than  y". 
Accordingly,  we  subtract  (1)  from  (2),  and  (2)  from  (3), 
and  by  comparison  obtain 


y 


x"  —  x' 


(4)- 


Since  the  form  of  the  locus  we  are  seeking,  whatever 
it  be,  is  (Art.  64)  independent  of  the  axes,  let  us  for 
convenience   refer  the  equation  to 
rectangular    axes,    OX    and    OY. 
Draw  the  indefinite  curve  AB,  to 
represent   for    the   time   being  the 
unknown  locus.     Take  P,  P",  P" 
as  the  three  points  x'y',  x"y",  x'"y'"; 
let  fall  the  corresponding  ordinates 
PM',  P"M",    P"M"';    draw   the 

chords  P'P",  P"P";  and  make  PR,  P'S  parallel  to 
OX.     Then,  from  (4),  we  have 

P'R      P"S 


that  is,  (Trig.,  818,) 

tan  P"P',K  =  tan  P"'P"#  .-.    P"PfR=P'"P'S. 

Hence,   the  three  points   Pr,  P",  P"  lie    on  one  right 
line. 

But  P',  P",  P"  are  any  three  points  of  the  locus.    P" 
may  therefore  be  anywhere  on  it  between  P'  and  P//r,  and 


RIGHT  LINE  IN  STRICT  ANALYSIS.  89 

is  independent  *  of  them.  Hence,  as  we  may  take  the 
points  as  near  each  other  as  we  please,  all  the  points  of 
the  locus  lie  on  one  right  line  ;  that  is,  the  locus  itself  is 
a  right  line. 

87.  A  third  proof  of  the  same  proposition  is  furnished 
by  transformation  of  co-ordinates. 


being  given  for  geometric  interpretation,  is  of  course 
referred  to  some  bilinear  system  of  co-ordinates.  Sup- 
pose the  original  axes  to  be  rectangular,  and  let  us 
transform  the  equation  to  a  new  rectangular  system 
having  the  origin  at  the  point  x'yr. 

To  effect  this,  write  (Art.  56,  Cor.  3,  cf.  Art.  58)  for 
x  and  y  in  the  given  equation  x'  -j-  x  cos  0  —  y  sin  6 
and  y'  -j-  x  sin  6  -f-  y  cos  6.  This  gives  us 

(A  cos  0  -f  B  sin  0}  x  —  (A  sin  6  —  B  cos  B)y  -f  Ax'  -f  £/  -f  C=  0 

as  the  equation  to  the  unknown  locus,  referred  to  the 
new  axes. 

Since  a/,  ?/,  and  0  are  arbitrary  constants,  we  may 
subject  them  to  any  conditions  we  please.  Let  us  then 
suppose  that  xf  and  y'  satisfy  the  relation 


and  that  the  value  of  6  is  such  that 

A  cos  6  -f-  B  sin  6  —  0  i.  e.  tan  0  =  — 


••:~  This  is  essential  to  the  argument.  For  three 
points  of  a  curve  may  lie  on  one  right  line,  if  the 
third  is  determined  by  the  other  two.  Thus,  in  the 
annexed  diagram,  P ,  P' ,  P"  are  three  points  on 
the  curve  AB ;  yet  they  all  lie  on  the  right  line 
P P" .  The  reason  is,  that  P"  is  determined  by 
joining  P  and  P". 


90  ANALYTIC  GEOMETRY. 

The  first  of  these  suppositions  means  that  the  new  origin 
is  taken  somewhere  on  the  unknown  locus,  since  its  co- 
ordinates satisfy  the  given  equation;  the  second,  that 
the  new  axis  of  x  makes  with  the  old,  an  angle  whose 

A 

tangent  is  --  -^  • 

Applying  these  suppositions  to  the  transformed  equa- 
tion, we  obtain,  after  reductions, 


Hence  (Art.  49,  Cor.  1)  the  locus  coincides  with  the  new 
axis  of  x. 

And,  in  general,  since  the  equation  Ax  -\-  By  -f  0=0 
can,  upon  the  suppositions  above  made,  always  be  reduced 
to  the  form  y  =  0,  we  conclude  that  it  represents  a  right 
line,  which  passes  through  the  arbitrary  point  x'y',  and 
makes  with  the  primitive  axis  of  x  an  angle  whose  tangent 
is  found  by  taking  the  negative  of  the  ratio  between  the 
co-efficients  of  x  and  y.  That  is  to  say,  since  these  co- 
efficients are  also  arbitrary,  Ax  -j-  By  -f-  C  =  0  is  the 
Equation  to  any  Right  Line. 

88.  We  have  thus  shown,  by  three  independent  demon- 
strations, that  we  can  take  the  empty  form  of  the  General 
Equation  of  the  First  Degree,  and,  merely  granting  that 
it  is  to  be  interpreted  according  to  the  convention  of  co- 
ordinates, evoke  from  it  the  figure  which  it  represents. 
It  must  not  be  supposed,  however,  that  we  ivere  ignorant 
of  the  figure  of  the  Right  Line  when  we  set  out  upon  the 
foregoing  transformations.  On  the  contrary,  each  of  the 
three  demonstrations  just  given  presupposes  the  figure 
of  the  Right  Line,  and  certain  of  its  properties.  What 
we  did  not  know  is,  that  the  equation  Ax  -f-  By  -)-  C=  0 
represents,  and  always  represents,  that  figure. 


EQUATION  OF  THE  FIRST  DEGREE.  91 

It  is  important  to  call  attention  to  this,  because  the 
significance  of  the  result  just  obtained  is  sometimes  over- 
estimated; and  because  the  case  of  the  Right  Line  is 
different  in  this  respect  from  that  of  any  higher  locus. 
In  the  strictly  analytic  investigation  of  loci  of  higher 
orders  than  the  First,  not  even  the  figure  of  the  curves 
is  presupposed,  but  is  conceived  as  being  learned  for  the 
first  time  from  their  equations.  But  the  whole  scheme 
of  Analytic  Geometry  takes  the  figure  and  elementary 
properties  of  the  Right  Line  for  granted  ;  as  is  obvious 
from  the  nature  of  the  convention  of  co-ordinates  and 
of  the  theorems  for  transformation. 

89.  Starting,  then,  from  Ax  -\-  By  -\-  C  —  0  as  the 
equation  to  the  Right  Line  in  its  most  general  form,  our 
next  step  will  naturally  be  to  determine  the  meaning 
of  the  constants  A,  _B,  and  C.  This  meaning  will  be 
found  to  vary  according  to  the  data  by  which  we  may 
suppose  the  position  of  a  right  line  to  be  fixed.  In  dis- 
cussing the  Right  Line  from  the  geometric  point  of  view, 
we  found  that  its  equation  assumed  three  forms,  de- 
pending upon  the  three  sets  into  which  the  data  for  its 
position  naturally  fall.  We  shall  now  see  that  the  gen- 
eral equation 


will  assume  one  or  another  of  those  forms,  according  as 
the  constants  in  it  are  interpreted  by  one  or  another  of 
the  sets  of  data. 

OO.  The  first  step  toward  a  correct  interpretation  of 
these,  is  to  observe  that  the  arbitrary  constants  in  our 
equation  are  but  two:  —  a  proposition  which  we  might 
infer  from  the  fact  that  in  an  equation  we  are  concerned 
only  with  the  mutual  ratios  of  the  co-efficients.  But 


92  ANALYTIC  GEOMETRY. 

its  truth  will  be  obvious,  if  we  consider  that  an  equa- 
tion may  be  divided  by  any  constant  without  affecting 
the  relation  between  its  variables,  and  therefore  without 
affecting  the  locus  which  it  represents.  Accordingly,  if 
we  divide  Ax  -f-  By  +  C  =  0  by  either  of  its  constants, 
for  example  (7,  we  obtain 

A         B 


a   form    in    which   there   are    only   two    arbitrary   con- 
stants. 

Corollary. — Hence,  Two  conditions  determine  a  right 
line.  Conversely,  A  right  line  may  be  made  to  satisfy  any 
two  conditions.  This  agrees  with  the  fact  that  the  data 
upon  which  the  three  forms  of  the  equation  to  the  Right 
Line  were  developed  geometrically,  are  taken  by  twos. 
(See  Arts.  78,  79,  80.) 

91.  We  now  proceed  to  the  analytic  deduction  of  those 
forms.  In  this  process,  the  meaning  of  the  ratios  among 
the  constants  A,  B,  and  C  will  duly  appear. 

I.  Let  the  data  be  the  angle  tvhich  the  line  makes  with 
the  axis  of  x,  and  its  intercept  on  the  axis  of  y.  We  use 
the  symbols  «,  o»,  m,  b  to  denote  the  same  quantities  as 
in  Art.  79. 

If  in  Ax  -f  By  +  C  =  0  we  make  y  =  0,  we  obtain 

„  /  A  «f   o  o\  (~\  rn  /-i  \ 

x  •=. ~r  —  (^fi.ri.  ooj  \j ±          \  /• 

If  we  make  x  =  0,  we  obtain 

y  =  —  -g=  (Art.  83)  OD  =  b  (2). 

OD      A 

Dividing  (2)  by  (1),  7yf,=  ^>. 


OD      A  ~7o-^.       rV 

/  \ 


MEANING  OF  THE  CONSTANTS.  93 

Hence,  (see  Trig.,  867,)    2  =  rin(«-a,)=~W    (3)' 

c 

From  (2),  we  have  B  =  —  -r ;  and  from  (3),  A  —  —  mB  — 

? —     Substituting  these  values  in  the  original  equation, 
b 

we  obtain 

mC         0 


.  • .     y  =  mx  -f-  b. 

A  C 

Corollary  1.— By  (3),  m  =  —  ^;  and  by  (2),  b=—^. 

Hence,  in  the  equation  to  a  given  right  line  the  ratio 
between  the  co-efficients  of  x  and  y,  taken  with  a  con- 
trary sign,  denotes  the  ratio  between  the  sines  of  the  angles 
which  the  line  makes  with  the  two  axes:  or,  when  the  axes 
are  rectangular,  it  denotes  the  tangent  of  the  angle  made 
with  the  axis  of  x ;  and  the  ratio  of  the  absolute  term  to 
the  co-efficient  of  y,  taken  with  a  contrary  sign,  denotes 
the  intercept  of  the  line  on  the  axis  of  y. 

Corollary  2, — Hence,  to  reduce  an  equation  in  the 
form  Ax  -j-  By  +  C  —  0  to  the  form  y  =  mx  -j-  6,  we 
merely  solve  the  equation  for  y. 

II.  Let  the  data  be  the  intercepts  of  the  line  on  the  two 
axes.  Here  a  and  b  have  their  usual  signification. 

Making  y  =  0  in  Ax  -f  By  -j-  0  =  0,  we  find  as 
before 

C  ^    /^ 

~~A~  ~a         (*)• 

^P 

Making  x  =  0,  we  obtain 

<7_ 


94 


ANALYTIC  GEOMETRY. 


c  c 

From  (1),  A  =  —  —  ;  and  from  (2),  JB  =  —  •  j  .      Substi- 

tuting for  A  and  B  in  the  equation,  and  reducing, 

x      y 
--f  f  =  1. 

a   '   6 

Corollary  1,  —  From  (1)  and  (2),  we  see  that  the  ratios 
of  the  absolute  term  to  the  co-efficients  of  x  and  y  re- 
spectively, taken  with  contrary  signs,  denote  the  intercepts 
of  the  line  on  the  axis  of  x  and  the  axis  of  y. 

Corollary  2.  —  To  reduce  Ax  -f  By  -f  C  =  0  to  the 


rm~  +  T-=slj  we  divide  it  by  its  absolute  term,  and, 
if  necessary,  change  its  signs. 

III.  Let  the  data  be  the  perpendicular  from  the  origin 
on  the  line,  and  the  angle  of  that  perpendicular  with  the  axis 
of  x.  We  use  p  and  a  in  the  same  sense  as  in  Art.  80. 

Making  y  and  x  successively  equal  to  0  in  the  general 
equation,  we  obtain,  as  before, 


(1), 


_      =  0l>=  (Trig.,  859)  (2). 


Ccosa 
From  (1),    A=  ---  — 


From  (2),    B  =  - 


C  cos 


Substituting    for    A    and    B    in 
the  general  equation,  we  have 


C  cos  a 


C  cos  ((o  —  « 


x  cos  «  -f~  ?/  cos  (co  —  «)  —  p. 


CONSTANTS  IN  TERMS  OF  DIRECTION-COSINES.  95 

Corollary,—  From  (1)  and  (2)  we  learn  that 

A  cos  a. 

-B       cos  (co  —  a)  ' 

or,  the  ratio  between  the  co-efficients  of  x  and  y,  denotes 
the  ratio  between  the  direction-cosines. 

Remark.—  The  reduction  of  Ax  -f  By  -f  C  =  0  to  the 
form  x  cos  a+y  cos  (to  —  a)=p  is  of  such  importance 
that  we  shall  discuss  its  method  in  a  separate  article. 

92.  Reduction  of  Ax  -f  By  -f  C  -=  0  to  the  form 

o?cos«-f  ycos(w  —  a)  =/>.  —  The  problem  may  be  more 
precisely  stated:  To  find  the  values  of  cos  «,  cos  (w  —  a), 
and  p,  w  ferm*  o/  A,  B,  and  C. 

If,  in  the  preceding  article,  we  divide  (1)  by  (2),  we 

find  5  =      ^co"'^      That  is>  (TriS'>  845>  IV  ;  839>) 

B 

—i  =  cos  co  -\-  sin  co  tan  a. 

B  — 

.  *  .    tan  a= 


= 


sin  co 

A  sin  co 


T/(l+tan2a)  i/(^l2  +^2—  2  ^l^cos  01) 

jB  cos  a  B  sin  w 

~~ 


C  cos  a  Csm 


co 


''      -  A  "V(A*  +B2  —  2  AB  cos  o>)  ' 

Therefore,  to  make  the  required  reduction,  Multiply  the 

..       n         7,7     _  sin  co  _    * 
equation  throughout  ly  ^/  (A2  +  B2  _  2  AB  cos  co)  ' 

*  The  following  elegant  solution  of  this  problem  is  by  SALMON  :  Conic 
Sections,  p.  20  : 

"  Suppose  that  the  given  equation  when  multiplied  by  a  certain  factor 
R  is  reduced  to  the  required  form,  then  RA  =  cos  a,  .K#  =  cos  j3.  But  it 
can  easily  be  proved  that,  if  a  and  ft  be  any  two  angles  whose  sum  is  w, 
we  shall  have 


9G  ANALYTIC   GEOMETRY. 

Remark.  —  Since  we  have  agreed  always  to  consider  p 
a  positive  quantity,  it  may  be  necessary  to  change  the  signs 
of  the  given  equation,  before  multiplying,  so  that  its  abso- 
lute term  when  transposed  to  the  second  member  may  be 
positive. 

Corollary  1,  —  If   the   axes   are   rectangular,   we   shall 

have 

A  B 

.Bv 

c 


Accordingly,  Ax  -f-  By  -j-  C  =  0  is  reduced  to  the  form 
x  cos  oi-\-y  sin  a  =  p,  by  dividing  all  its  terms  by  V  Az-\-B2, 
after  the  necessary  changes  of  sign. 

Great  importance  pertains  to  the  transformation  ex- 
plained in  this  corollary.  The  general  reduction  to 
x  cos  a  -f-  y  cos  (to  —  a)=p  is  of  comparatively  infre- 
quent use. 

Corollary  2.  —  From  the  values  of  p  above  obtained,  we 
learn  that  the  length  of  the  perpendicular  from  the  origin 
upon  the  line  Ax  +  By  -|-  C=  0  is 


cos2  a  +  cos2  (3  —  2  cos  a  cos  (3  cos  u  =  sin2  a). 

Hence,  B2  (A*+  B'2—  2  AB  cos  u)  =sin2w;   and  the  equation  reduced  to 
the  required  form  is 

A  sino  JBsincj  _  ^  ^ 

Bz-2  AB  cos  u)J    \ 


And  we  learn  that 


B*  -  2  ABcos  u)  '      y(A*  +  B2  -  2  AB  cos  u) 

are  respectively  the  cosines  of  the  angles  that  the  perpendicular  from  the 
origin  on  the  line  Ax-\-By-{-  C=Q  makes  with  the  axes  of  x  and  y;  and 

that         *         t-  is  the  length  of  that  PerPendicular-" 


POL  A  R  EQ  UA  TION  B  Y  ANAL  YSIS.  97 

C  sin  co 
~  ]/  (A2  +  B2—  2  AB  cos  at)  ' 

and  that  this  length  becomes 

C 

V(A*  +  B>)  » 

when  the  axes  are  rectangular. 

93.  Polar  Equation  to  the  Right  Line,  deduced 
analytically.  —  The  polar  equation  may  be  obtained 
from  the  Cartesian  as  follows  :  Let  Ax  -f-  By  -f  C  =  0, 
referred  to  rectangular  axes,  be  reduced  to  the  form 

x  cos  o.-\-y  sin  a=p. 

Transforming  to  polar  co-ordinates,  we  have  (Art.  57, 
Cor.)  x  =  p  cos  $,  and  y  =  p  sin  0  :  and  the  equation 
becomes 

/>  (cos  6  cos  «  -f  sin  d  sin  a)  =_p  ; 

that  is,  jO  cos  (6  —  a)  =  p, 

the  equation  of  Art.  82. 

Corollary.  —  Transforming  the  original  equation  to 
polars,  we  have 

p  (A  cos  6  +  B  sin  6)  +  0=  0. 


Hence,  an  equation  in  the  form  p  (A  cos  d  +  J5sin  0)  =(7 
may  be  reduced  to  the  form  p  cos  (0  —  a)=p  by  dividing 
each  term  by 


94.  Before  advancing  to  the  more  particular  forms 
of  the  equation  to  the  Right  Line,  the  student  should 
make  sure  of  having  mastered  the  general  ones  which 
precede,  and  the  principles  which  have  been  developed  in 
the  course  of  the  discussions  just  closed.  To  this  end, 
let  the  following  exercises  be  performed. 


98  ANALYTIC  GEOMETRY. 

EXAMPLES. 

1.  Transform  3x  —  5y  +  6  =  0  to  y  =  0.     What  is  the  angle 
made  by  the  new  axis  of  x  with  the  old  ? 

2.  What  angle  does  the  line  Sx  +  12y  -f  2  =  0  make  with  the 
axis  of  x  f     What  is  the  length  of  its  intercept  on  the  axis  of  y  1 
In  which  of  the  four  angles  does  it  lie  ? 

3.  Find  the  intercepts  of  the  line  x  -f-  3?/  —  3  =  0. 

4.  Find  the  ratio  between  the  direction-cosines  of  the  right  line 
2x  +  3^  +  4  =  0,  «  being  =  60°. 

5.  What  is  the  tangent  of  the  angle  made  with  the  axis  of  x 
by  the  perpendicular  from  the  origin  on  3x  —  1y  —  6  =  0? 

6.  Reduce  all  the  equations  in  the  previous   examples  to  the 
form  x  cos  a,  -\-y  sin  a—  p,  and  determine  a  and  p  for  each  line. 

7.  Reduce  'Sx  +  4y  =  12  to  the  form  x  cos  a  +  y  cos  (w  —  a)  —  p. 
What  are  the  values  of  a  and  (w  —  a),  supposing  a  successively 
equal  to  30°,  45°,  60°,  and  sin"1  f  ? 

8.  Find  the  length  of  the  perpendicular  from  the   origin  on 
3x  +  4y  +  12  =  0,  under  the  several  values  of  o  last  supposed. 

9.  Find   the   length  of  the  perpendicular  from  the   origin   on 
3x  — 4y  —  12=0,  axes  being  rectangular. 

10.   Reduce  2  p  cos  6  —  3  p  sin0  =  5  to  the  form  p  cos  (6  —  a)=p. 
What  are  the  values  of  a  and  p  ? 

THE    RIGHT    LINE    UNDER   SPECIAL    CONDITIONS. 

95.  Equation  to  the  right  line  passing  through 
Two  Fixed  Points. — Let  the  two  points  be  xfyf,  x"y". 

Since  they  are  points  on  a  right  line,  their  co-ordinates 
must  satisfy  the  equation 

Ax  +  By  +  0=  0  (1), 
and  we  therefore  have 

Ax?  +By>  +C=Q  (2), 

Ax"  +  By" +0=0  (3). 


RIGHT  LINE  THROUGH  TWO  POINTS.  99 

Subtracting  (2)  from  (1)  and  (3)  successively,  we  obtain 

A  (x  —  xf)  -j-  B  (y  —  yf)  =  0, 
A(x"-x')+£(y"-1J')=0, 
and  thence 

y   —   yf  y't     —     yf 

y_ &    y y  /A\ 

x—x'  ~  x"  —  xf 

which  is  the  equation  required.  For  it  is  the  equation 
to  some  right  line,  since  it  is  of  the  first  degree;  and  it 
is  the  equation  to  the  line  passing  through  the  two  given 
points,  because  it  vanishes  when  either  x'  and  y'  or  x" 

and  y"  are  substituted  in  it  for  x  and  y. 

&  tj 

Corollary  1. — Equation  (4)  may  evidently  be  written 
(yf  —  y"}  x  —  (xf  —  x")  y  +  x'y"  —  y'x"  =  0  : 

a  form  often  useful,  though  the  form  (4)  is  more  easily 
remembered. 

Corollary  2. — If  in  the  last  equation  we  suppose  x"=  0 
and  y"=  0,  we  obtain 


the  equation  to  the  right  line  passing  through  a  fixed  point 
and  the  origin. 

Remark  —  The  same  equation,  (4),  might  have  been  obtained 
geometrically.  For,  since  the  triangles  PRP',  P'tSP"  are  similar, 
we  have 

PR   _P'S 

RP/  ~  SP"  ' 

that  is,  after  changing  the  signs  of  the 
equation, 

y  -  y'      y"-y' 


O    M    M'   M" 


100 


ANALYTIC  GEOMETRY. 


It  is  worth  while  to  place  the  analytic  and  geometric  proofs  thus 
side  by  side,  in  order  to  make  sure  that  the  geometric  meaning  of 
all  the  symbols  in  the  equations  developed,  shall  be  clearly  under- 
stood. 

96.  Angle  between  two  right  lines  given  by 
their  equations.  —  All  formulae  for  angles  are  greatly 
simplified  by  the  use  of  rectangular  axes.  We  therefore 
present  the  subject  of  this  article  first  in  the  form  which 
those  axes  determine. 

Let  the  two  lines  be  y  =  mx  +  b 
and  y  =  m'x  -f-  bf  ;  and  let  tp  =  the 
angle  between  them.  From  the  dia- 
gram, <p  =  a!  —  a  ;  and  we  have 
(Trig.,  845,  vi) 


tan  a'  —  tan  a 


.  78,  Cor.  1). 


Corollary  1.  —  If  the  equations  were  given  in  the  form 
Ax  +  By  +  C=Q,  A'x  +  B'y  -f  O  =  0,  we  should  have 

A.  A! 

(Art.  91,  1,  Cor.  1)  m  =    -  -g  ,  and  m'  =  —  -™  .    Hence, 

in  that  case, 

AB'—A'B 
*~- 


Corollary  2  —  If  <p  =  0  or  ?r,  tan  tp  =  0  ;  and  we  have 

m1  —  m  =  Q  or  ABf  —  A'B  =  0: 
the  condition  that  two  right  lines  shall  be  parallel. 
Corollary  3.  —  If  tp  =  90°,  tan  <p  =  oo;  and  we  have 
1  +  mm'  =  0  or  A  A'  +  BB'  =  0  : 

the  condition  that  two  right  lines  shall  be  mutually  perpen- 
dicular. 


ANGLE  BETWEEN  TWO  EIGHT  LINES.          101 

OT«    Angle  between  two  lines,  axes  being-  oblique. — From 
Art.  92  we  shall  have  cos  a  =  A  B  cos  ^  >  and 


.     Hence  (Trig,  838) 


'(A''2  +  Eri  —  2  A'B'  cos  w) 


sin  a   = 


!/  (^4/2  +  £/2  -  2  ^S7  cos  w)  ' 
B  —  A  cos  w 


Therefore  (Trig,  845,  HI  and  iv 
.    ,_,_*' 


x_  —  (AB/  +  ^x^)  cos  c^ 

~ 


Whence  (Trig,  839) 

—  .d'.B)sino 


—  (AB/  +  ^x£)  cos  w  ' 
a  formula  which  evidently  becomes  that  of  Art.  96,  Cor.  1,  if  a  —  90°. 

Corollary  1 — The  condition  that  the  two  lines  shall  be  parallel, 
is 

AB'  —  A'B  =  0 : 

from  the  identity  of  which  with  the  condition  of  Art.  96,  Cor.  2, 
we  learn  that  the  condition  of  parallelism  is  independent  of  the  value 
of  w.  The  same  follows  from  the  fact  that  the  condition  itself  is 
not  a  function  of  w. 

Corollary  2. — The  condition  that  the  lines  shall  be  perpendicular 
to  each  other,  is 

AA'  +  B&  —  (AB'  +  A'B)  cos  w  =  0. 

98.  Equation  to  a  right  line  parallel  to  a  given 
one. — Let  the  given  line  be  y  =  mx  -\-  b.  The  required 
equation  will  be  of  the  form 

y  —  m'x  -\-  b'. 


102  ANALYTIC  GEOMETRY. 

But  in  this,  the  condition  of  parallelism  (Art.  96,  Cor.  2  ; 
Art.  97,  Cor.  1)  gives  us  mf  =  m.  The  required  equation 
is  therefore 

y  =  mx  -f-  b'. 

Corollary,  —  If  the  given  line  were  Ax  -\-  By  -f-  C=  0, 
the  equation  would  be  of  the  form  Ax  -\-  B'y  -\-  C'  =  0. 

A  Cf 

In  that  case,  m  =  —  j^  ,  and  bf  =  —  IT,  .     The  required 

BC' 

equation  would  therefore  be,  after  writing  C"  for  -/-  , 


From  this  we  infer  that  the  equations  to  parallel  right  lines 
differ  only  in  their  constant  terms. 

99.  Rectangular  equation  to  a  right  line  per- 
pendicular to  a  given  one.  —  The  equation  will  be  of 
the  form  y  =  m'x  +  6',  in  which  m'  is  determined  by  the 
condition  (Art.  96,  Cor.  3) 

1  +  mm'  =  0. 
Hence,  m!  =  —  —  ;  and  the  equation  is 

y  =  —  —  x  +  bf. 
y  m 

Corollary,  —  When  the  given  line  is  Ax  -f-  By  +  0=  0, 
the  equation  to  its  perpendicular  is  (Art.  91,  I,  Cor.  1) 

Ay  —  Bx  +  Ci=Q. 

Hence,  if  two  right  lines  are  perpendicular  to  each  other, 
their  rectangular  equations  interchange  the  co-efficients  of 
x  and  y,  and  change  the  sign  of  one  of  them. 


SYMMETRIC  EQUATION  TO  RIGHT  LINE.       103 

100.  Equation  to  a  right  line  perpendicular  to  a  given 
one,  axes  being  oblique. — From  the  condition  of  perpendicularity, 
(Art.  97,  Cor.  2,) 

A'  (A  —  B  cos  w)  +  B'  (B  —  A  cos  w)  =  0 

A'      B  —  A  cos  w 
Hence'  -W  =  A-Bw*»> 

and  the  required  equation  (Art.  91,  I,  Cor.  1)  is 

(A  —  B  cos  w)  y  —  (B  —  A  cos  «)  x  +  C2  =  0. 

101.  Equation  to   a   right   line  parallel   to   a 
given    one,   passing   through    a    Fixed   Point. — By 

Art.  98  we  have  the  form  of  the  equation, 

y  =  mxjr  V  (1). 

Calling  the  fixed  point  x'y',  we  therefore  obtain 

y'  =  mxf  +  ^  : 

bf  =  y'  —  mxf. 

Substituting  for  bf  in  (1),  the  equation  sought  is 
y  —  yf=m(x  —  x'}. 

Corollary  1. — If  in  this  equation  we  suppose  m  indeter- 
minate, the  direction  of  the  line  is  indeterminate ;  and  we 
have  the  equation  to  any  right  line  passing  through  a  fixed 
point. 

Corollary  2 — Let  sin  a  :  sin  to  =  k,  and  sin  (w  —  «)  : 
sin  co  =  h.  Then  k  :  h  -\-  sin  a  :  sin  (co  —  a)  =m ;  and 
the  equation  may  be  written. 


k  h 

or,  if  we  denote  either  of  these  equal  ratios  by  Z, 

y    y'  ___  x~  x'  _  j . 

An.  Ge.  12. 


104  ANALYTIC   GEOMETRY. 

a  formula  often  convenient,  and  known  as  the  Symmetrical 
Equation  to  the  Right  Line. 

Corollary  3. — If  the  axes  are  rectangular,  we  have 
k  :  h  =  sin  a  :  cos  a ;  and  the  equation  may  be  more 
conveniently  written 

y— yf      x  —  xr 


where  s  and  c  are  abbreviations  for  the  sine  and  cosine 
of  the  angle  which  the  line  makes  with  the  axis  of  x. 

1O2.  Geometric  meaning  of  the  ratio  /. — If,  in 

the  annexed  diagram,  P'  denote  the  fixed  point  x'y' ,  and 
P  any  point  of  a  right  line 
passing  through  it,  PR  being  /Y 

parallel  to  OX,  \  / 


y—y' 

PR  sin  co                        /  \p 

1 

\ 

k 

smPP'R  >                   /    R/..\P 
P'  R  sin  co                  / 

h 

But  (Trig., 
Hence, 

sin  P'PR             /°     M     M' 
871) 
PR  sin  co                    P'Rsmco 

T\X 

siuPP'R      1  1       sin  P'PR  ' 

y  —  y'      x—x' 

k              h          l 

denotes  the  distance  from  a  fixed  point  x'y'  to  any  point 
xy  of  a  right  line  passing  through  it. 

Remark  1. — So  long  as  the  point  xy  is  variable,  I  is 
of  course  indeterminate.  But  the  formula  enables  us  to 
find  the  distance  from  x'y'  to  any  given  point  on  the  line, 
by  merely  substituting  for  x  or  y  the  abscissa  or  ordinate 
of  such  given  point. 


LINE  CUTTING  ANOTHER  AT  GIVEN  ANGLE.  105 

Thus,  supposing  the  given  point  of  the  line  to  be  x"yff , 
we  should  have  (assuming  the  axes  for  convenience  to  be 
rectangular) 

y"-yf 
i       zs' 

x"  -  x' 

~r  =c- 

Squaring   both    sides    of  these    equations,    adding,    and 
remembering  (Trig.,  838)  that  s2  -f  c2  =  1,  we  obtain 


which  agrees  with  the  formula  (Art.  51,  I,  Cor.  1)  for 
the  distance  betiveen  two  given  points. 

Remark  2. — The  signs  -\-  and  —  in  connection  with  I 
denote  distances  measured  along  the  line  in  opposite  di- 
rections from  xryf.  Thus,  if  -f  I  were  measured  in  the 
direction  P'D,  —  I  would  be  measured  in  the  direction 
P'  T;  and  vice  versa.  Speaking  with  entire  generality, 
+  I  must  be  laid  off  from  x'y'  in  the  positive  direction 
of  the  line.  (Aft.  50,  Note),  and  —  I  in  the  negative. 

1O3.  Rectangular  equation  to  a  right  line 
passing  through  a  Fixed  Point,  and  cutting  a 
given  line  at  a  Given  Angle. — Let  the  given  line  be 
y  =  mx  -f  5,  and  let  6  =  the  given  angle.  The  equation 
sought  (Art.  101,  Cor.  1)  is  of  the  form 

</-/=™'(*-O, 

in  which  m'  is  to  be  determined  from  the  conditions  of  the 
problem. 

The  line  in  question  obviously  makes  with  the  axis  of  x 
an  angle  u!  =  a  -|-  6.  Hence,  (Trig.,  845,  v,) 

m  -f-  tan  6 
1  —  m  tan  d 


106  ANALYTIC  GEOMETRY. 

Substituting  this  value  for  mr,  the  required  equation  is 
m  +  tanfl 


Corollary  1.  —  Dividing  both  terms  of  m1  by  tan  6,  this 
equation  may  be  written 

m  cot  6  -f-  1  , 


Accordingly,  if  0  =  90°,  it  becomes 

y  —  #'  =  -^0—  O: 

the  equation  to  a  right  line  passing  through  a  fixed  point, 
and  perpendicular  to  a  given  line. 

Corollary  2.  —  By  substituting  for  —  m  its  value  A  :  B, 
we  obtain 

^(y-»0  -5(*-*0=o 

as  the  equation  to  the  perpendicular  of  .Ar-f-JBy  +(7=0, 

passing  through  aty'  :  the  co-efficients  of  which  evidently 
satisfy  the  criterion  of  the  corollary  to  Art.  99. 

1O-4.    Equation  to  a  right  line  passing:  through  a  Fixed 
Point  and  cutting  a  given  line  at  a  Given  Angle,  axes  being 
oblique.  —  Let  the  given  line  be  Ax  -{-By  -f-  C=  0.    Since  a'=  a  -j-  0, 
' 


—    . 
By  putting  for  m'  its  value,  the  required  equation  assumes  the 


form 


From  Art.  97,  we  have 


(AB/  —  A' 

~ 


A  A'  +  BE'  —  (AB/  +  A'B}  cos  «  ' 
therefore 

A/  _  A  sin  w  —  (B  —  A  cos  «)  tan  0 
W  ~~  B  sin  «  +  (A  —  B  cos  «)  tan  fl  ' 


LENGTH  OF  A  PERPENDICULAR. 


107 


and  the  required  equation  is 


{A  sin  u  —  (B  —  A  cos  w)  tan  6}  (x  —  x/) 
—  5  cosu)  tan  0}  (y  — 


/)  \ 
)  J    = 


Corollary  1  —  When  the  given  line  is  y  =  ma;  +  6,  the  equation 
assumes  the  form 

,  _  m  sin  «  -f-  (1  -f-  m  cos  w)  tan  0  . 

sin  w  —  (m  +  cos  w)  tan  #  '  *  ~~     '' 

This  evidently  becomes  the  equation  of  Art.  103  when  u  =  90°. 

Corollary  2  —  Dividing  the   equation  by  tan    6,    and    supposing 

9  =  90°,  we  obtain 

(B  —  Acosu)  (x  —  x/)  —  (A  —  Bcos^  (y  —  /)  =  0: 

the  general  equation  to  the  perpendicular  of  a  given  line,  passing  through 
n  fixed  point. 

This  might  have  been  obtained  directly  from  the  equation  of 
Art.  100. 

1O5.  Length  of  the  perpendicular  from  any 
point  to  a  given  right  line.  —  Let  the  given  point  be 
xy,  and  the  given  line  x  cos  a-\-y  cos  (co  —  a)  —  p  =  0. 

Represent  the  given  point  at  P, 
and  the  given  line  by  DT.  Let 
PQ  be  the  required  perpendicular. 
Draw  PM  parallel  to  OD,  OS  par- 
allel to  PQ,  and  PAS;  MN  parallel 
to  DT.  Then, 

=  OMcos  MON=  x  cos  « 


vP 


a)  : 


In  the  foregoing  discussion,  the  point  P  and  the  origin 
were  assumed  to  be  on  opposite  sides  of  the  given  line  ; 
had  they  been  supposed  on  the  same  side,  OR  would  have 
been  greater  than  OS,  and  we  should  have  had 


PQ=  OR  — 


=  p  —  x  cos  «  — 


CQ§  (ct 

l( 


~  a). 


108  ANALYTIC   GEOMETRY. 

Hence,  if  a  perpendicular  be  let  fall  from  any  point  xy  to 
the  line 

x  cos  a  -f-  y  cos  (to  —  «)  — p  =  0, 

its  length  will  be 

db  (x  cos  a -\- y  cos  (w  —  a)  — p) 

according  as  the  point  and  the  origin  lie  on  opposite  sides 
of  the  line,  or  on  the  same  side. 

Remark — The  student  will  observe  that  we  here  use  the  same 
symbols  xy  to  denote  the  co-ordinates  of  the  point  from  which  the 
perpendicular  is  dropped,  and  those  of  any  point  on  the  given  line. 
But  it  must  not  be  supposed  that  the  xy  of  the  point  and  the  xy 
of  the  line  have  necessarily  the  same  values.  Generally,  of  course, 
they  have  not;  for  the  point  from  which  the  perpendicular  falls, 
is  generally  supposed  not  to  be  a  point  on  the  line.  It  may  be;  and 
when  it  is,  the  length  of  the  perpendicular  vanishes.  Hence,  sup- 
posing the  quantity  x  cos  a  +  y  cos  («  —  a)  — p  to  denote  the  length 
of  the  perpendicular  from  xy  on  a  given  line,  the  equation 

x  COS  a  -f-  y  cos  (w  —  a)  —  p  =  0 

signifies  that  the  point  lies  somewhere  on  the  line  in  question. 

The  double  use  of  xy  may  cause  some  confusion  at  first,  but  its 
advantages  more  than  compensate  for  the  attention  required  to 
overcome  this. 

Corollary  1. — If  the  axes  are  rectangular,  the  length 
of  the  perpendicular  is 

±  (x  cos  a  -f  y  sin  a  — p) 

according  as  the  point  and  the  origin  lie  on  opposite  sides 
of  the  line,  or  on  the  same  side. 

Corollary  2, — The  length  of  the  perpendicular  from  xy 
on  the  line  Ax  +  By  +  C=  0  (Art.  92  cf.  Cor.  1)  is 

(Ax  +  By  -f-  C)  sin  to  Ax+By+C 

~  y (A2  -}-  &  —  2  AB  cos  to)  c       :   y(A2+&)~ 

according  as  the  axes  are  oblique  or  rectangular. 


I 


POINT  OF  INTERSECTION.  109 


Corollary  3.  —  The  perpendicular  from  a  point  on  the 
same  side  of  a  line  as  .  the  origin  must  have  the  same 
sign  as  p.  But  we  have  agreed  (Art.  80,  Rem.)  that  p 
shall  always  be  positive;  and  we  have  seen  that  the  per- 
pendicular changes  sign  in  passing  from  one  side  of  the 
line  to  the  other.  Hence,  Perpendiculars  falling  on  the 
side  of  a  line  next  the  origin  are  positive  ;  and  those  falling 
on  the  side  remote,  are  negative. 

1O6.  To  find  the  point  of  intersection  of  two 
right  lines  given  by  their  equations.  —  Eliminate 
between  the  equations  to  the  two  lines  :  the  resulting 
values  of  x  and  y  (Art.  62)  are  the  co-ordinates  of  the 
required  point. 

Thus,  in  general,  the  lines  being  Ax  -f  By  -j-  C=  -  0 
and  A'x  +  B'y  -f  O  =  0,  we  have 

BC'—  B'Q  CA'  —  C'A 


A'B>      ~  ABf  —  A'B 
as  the  co-ordinates  of  the  common  point. 

1O7.  Equation  to  a  right  line  passing  through 
the  intersection  of  two  given  ones.  —  If  we  multiply 
the  equations  to  two  given  lines  each  by  an  arbitrary 
constant,  and  add  the  results,  thus  : 

I  (Ax  +By+CT)  +  m  (A'x  +  B'y  +  C'}  =  0, 

the  new  equation  will  represent  a  right  line  passing 
through  the  intersection  of  the  lines  Ax  +  By  -f-  0=  0 
and  A'x  +  B'y  +  (?  =  Q. 

For  it  manifestly  denotes  some  right  line,  since  it  is 
of  the  first  degree.  Moreover,  it  is  satisfied  by  any 
values  of  x  and  y  that  satisfy  Ax  +  By  -f-  (7=0  and 
A'x  -f  B'y  -f  Cf=  0  simultaneously  ;  for  its  left  member 
must  vanish  whenever  the  quantities  Ax  +  By  -j-  C  and 


110  ANALYTIC  GEOMETRY. 

A'x  -f-  B'y  -j-  C'  are  both  equal  to  zero.  That  is,  it 
passes  through  a  point  whose  co-ordinates  satisfy  the 
equations  of  both  the  given  lines.  But  such  a  point 
(Arts.  62,  106)  is  the  intersection  of  the  two  lines. 

Corollary. — By  varying  the  values  of  the  constants  I 
and  m,  we  can  cause  the  above  equation  to  represent 
as  many  different  lines  as  we  please,  all  passing  through 
the  intersection  of  the  two  given  ones. 

Remark. — The  truth  of  the  above  equation  is  inde- 
pendent of  a),  and  of  the  form  of  the  equations  to  the 
given  lines.  This  is  manifest  from  the  method  by  which 
it  was  obtained.  It  may  therefore,  when  convenience 
requires,  be  written 

I  {x  cos  o.-\-y  cos  (co  —  a)  — p  } 


A 

-\-m{x  cos/3  +  2/cos  (at  —  ft)  — p'}  ' 
or 
I  (x  cos  a  -j-  y  sin  «  —  p)  -\-  m  (x  cos  ft  -f-  y  sin  ft  — p'}  =  0. 

1O8.  Meaning  of  tne  equation  L+kI/=O.—If 

we  put  k  =  m  :  I,  and  represent  by  L  and  L'  the 
quantities  which  are  equated  to  zero  in  the  equations 
of  the  two  given  lines,  the  equation  of  the  preceding 
article  becomes 

L  +  kL'  =  0. 

We  shall  now  prove  that  this  is  the  equation  to  any  right 
line  passing  through  the  intersection  of  two  given  ones.  * 


*  The  beginner  may  suppose  that  this  has  been  done  already,  in  the 
preceding  article.  But  we  merely  proved  there,  that  the  equation  may 
represent  an  infinite  number  of  lines  answering  to  the  given  condition. 
Now,  that  an  infinite  number  of  lines  is  not  the  same  as  all  the  lines 
passing  through  the  intersection  of  two  others,  is  evident.  For  between 
two  intersecting  right  lines  there  are  two  angles,  supplemental  to  each 
other,  in  each  of  which  there  may  be  an  infinite  number  of  lines  passing 
through  the  common  point. 


LINE  THROUGH  INTERSECTION.  HI 

Let  6  =  the  angle  made  with  the  axis  of  x  by  the  line 
which  the  equation  represents.     Then  (Art.  91, 1,  Cor.  1) 

sin  0  A  +  kA' 


sin  (a*  —  0)  B  -f  kB'  ' 

Therefore 

(A  +  kAf)  sin  CD 


tan  u  = 


(A  cos a)—B)+k  (A'  cos  «*  —  B') 

Hence,  as  k  may  have  any  value  from  0  to  oo,  and 
be  either  positive  or  negative,  tan  6  may  have  any 
value,  either  positive  or  negative,  from  0  to  oo.  That 
is,  the  equation  is  consistent  with  any  value  of  6  what- 
ever. 

Therefore,  if  L  =  0  and  L'  =  0  are  the  equations  to 
two  right  lines, 

L  -f  kL'  =  0 

is  the  equation  to  any  right  line  passing  through  the 
intersection  of  the  two. 

Corollary  1. — The  equation  to  a  particular  line  inter- 
secting two  others  in  their  common  point,  is  formed 
from  the  above  by  assigning  to  k  such  a  value,  in  terms 
of  the  conditions  which  the  line  must  satisfy,  as  the 
relation  L  -f  kU  =  0  implies. 

Thus,  if  the  condition  were  that  the  intersecting  line 
make  with  the  axis  of  x  a  given  angle  —  0,  we  should 
have,  from  the  value  of  tan  6  above, 

A  sin  co  —  (A  cos  co  —  B  )  tan  6 
~  Z'~sin  «>  —  (Ar  "cos^i >~~ rF'Jtan^  ' 

or,  in  case  the  axes  were  rectangular, 

A  -f-  B  tan  0 
~A'  +  £'t&u0' 
An.  Ge.  13. 


112  ANALYTIC   GEOMETRY. 

If  the  condition  were  that  the  intersecting  line  pass 
through  a  fixed  point  x'y'  ,  we  should  have 

(Axf  +  By'  +  tf)  rM  (Ax>  +  B'y'  +  Cr)  =  0  : 

Ax?  +  3^+0 

A'x'  4-  B'y1  +  6"  ' 

Corollary  2.  —  The  si0w  of  k  has  a  most  important 
geometric  meaning.  Obviously,  in  order  to  change  its 
sign,  a  quantity  must  pass  through  the  value  0  or  oc. 
Now  if  k  —  0,  we  have  by  Cor.  1 

A 

A  +  B  tan  6  =  0  .  •  .  tan  0  =  —  -g-  . 

And  if  k  =  GO, 

A! 
A'  +  jB'  tan  0  =  0  .  •  .  tan  0  ==  —  -g?-. 

That  is,  (Art.  91,  I,  Cor.  1,)  a£  ^Ae  instant  when  k  changes 
sign,  the  line  which  passes  through  the  intersection  of  two 
given  ones  coincides  with  one  of  them.  Hence,  of  the  lines 
L  -j-  ~kL'  =  0  and  L  —  kL'  =  0,  one  lies  in  the  angle  be- 
tween Tj  =  Q  and  U  =  0  supplemental  to  that  in  which 
the  other  lies. 

It  now  remains  to  determine  which  of  these  supple- 
mental angles  corresponds  to  -f-  k,  and  which  to  —  k. 
If  the  two  lines  are  x  cos  a  -f-  y  cos  (co  —  «)  —  p  =  0  and 
—  /?)  —  £/—  0,  we  shall  have 

x  cos  a  -f  i/  cos  (w  —  «)  —  jy 


cos     +     cos   w  —      — 


that  is,  (Art.  105,)  one  of  the  geometric  meanings  of  k  is, 
the  negative  of  the  ratio  between  the  perpendiculars  let  fall 
on  tivo  given  lines  from  any  point  of  a  line  passing  through 
their  intersection.  Hence,  when  k  is  positive,  those  per- 
pendiculars have  unlike  signs  ;  and  Avhen  k  is  negative, 
their  signs  are  like.  That  is,  (Art.  105,  Cor.  3,)  the  per- 
pendiculars corresponding  to  +  k  fall  one  on  the  side  of 


CRITERION  OF  ITS  POSITION.  113 

one  line  next  the  origin,  and  the 
other  on  the  remote  side  of  the  other 
line  ;  while  those  corresponding  to 
—  k  fall  both  on  the  side  of  the  lines 
next  the  origin  or  both  on  the  side 
remote  from  it.  In  other  words,  the 
line  corresponding  to  +  k  lies  in  the 
angle  remote  from  the  origin,  e.  g.  P'L';  and  the  line  corre- 
sponding to  —  k  lies  in  the  same  angle  as  the  origin,  e.  g.  PL. 

For  convenience,  we  shall  call  the  angle  R'SQf,  remote 
from  the  origin,  the  external  angle  between  the  given 
lines  ;  and  the  angle  RSQ,  in  which  the  origin  lies,  the 
internal  angle. 

Hence,  if  L  =  0  and  Lf  =  0  are  the  equations  to  any 
two  right  lines, 

L  4-  kL'  =  0 

denotes  a  line  passing   through   their   intersection   and 
lying  in  the  external  angle  between  them  ;  and 

L  —  kL'  ==  0 
denotes  one  lying  in  the  internal  angle. 

1O9.  Equation  to  a  right  line  bisecting  the 
angle  between  two  given  ones.  —  Any  point  on  the 
bisector  being  equally  distant  from  the  two  given  lines, 
we  have  (Art.  105  cf.  Cor.  2) 

Ax  -f  By  +  0  A'x  +  B'y  +  Cf 


or 

co  —  ft)  —  pf}*  (2), 


*  The  student  may  at  first  think  that  both  members  of  (1)  and  (2) 
should  have  the  double  sign.  But  since  an  equation  always  implies  the 
possibility  of  changing  its  signs,  it  is  evident  that  we  should  write  the 
expressions  as  above. 


114  ANALYTIC  GEOMETRY. 

according  as  the  given  lines  are  Ax  +  By  -{-  (7— 0  and 
A'x  4-  B'y  -f-  (?'  =  0  or  a;  cos  a  +  y  cos  (w  —  a)  —  p  ==  0 
and  a;  cos  /9  +  #  cos  (w  —  /?)  —  p'  —  0.  Expressions  (1) 
and  (2)  are  the  principal  forms  of  the  required  equation. 

Corollary  1. — When  the  axes  are  rectangular,  the 
equation  becomes 

Ax  +  By+C  A'x  +  B'y+C' 

l/(A*+&)  V(A'*  +  B'*) 

or 

x  cos  a  -f-  y  sin  «  — JP  =  ±  (x  cos  /?  +  ?/  sin  /? — p')    (4). 

Corollary  2. — Expressions  (1),  (2),  (3),  (4)  are  evidently 
in  the  form  L  ±  kL'  =  0.  Hence  (Art.  108,  Cor.  2) 
there  are  two  bisectors :  one  lying  in  the  external  angle 
of  the  given  lines,  and  the  other  in  the  internal.  For 
the  sake  of  brevity,  we  shall  call  the  former  the  external 
bisector ;  and  the  latter,  the  internal  bisector. 

Corollary  3. — If  we  put,  as  we  conveniently  may, 

a  =  x  cos  a  -f-  y  cos  (co  —  «)  — p, 
fi  =  x  cos  ft  -\-  y  cos  (co  —  ft)  —  pf, 

expressions  (2)  and  (4)  will  be  included  in  the  brief  and 
striking  form 

a  ±0  =  0. 

Hence,  if  a  —  0,  ft  =  §  are  the  equations  to  any  two  right 
lines,  the  line 

a  -f-  ft  •=  0 

bisects  the  external  angle  between  them ;  and  the  line 

«-/3  =  0 
bisects  their  internal  angle. 


BISECTOR  OF  ANGLE.  115 

Caution.  —  From  the  proposition  just  reached,  the  stu- 
dent is  apt  to  rush  to  the  conclusion  that 

L  ±  L'  -  0 

is  the  equation  to  the  bisector  of  the  angle  between  the 
lines  L  ==  0  and  Lr  =  0,  without  regard  to  the  values  of 
L  and  U  .  This  is  a  grave  error.  It  assumes  that  the 
value  of  k,  in  the  case  of  a  bisector,  is  always  ±  1. 
When  the  equations  to  two  lines  are  in  terms  of  the 
perpendicular  from  the  origin  and  its  angle  with  the 
axis  of  x  ;  that  is,  when  L~x  cos  a  -f-  y  cos  (co  —  a)  —  p 
and  L'  —  x  cos  /9  -j-  y  cos  (co  —  /9)  —  pf,  k  =  ±  1.  But 
from  equation  (1)  we  have 


(A*  +  B'2  —  2  A'Bf  cos 


which  is  obviously  not  in  general  equal  to  ±  1.     The 
condition  that  it  shall  have  that  value  is 

A2  -f  B2  —  2  AB  cos  co  =  A'2  +  B'2  —  2  A'B'  cos  co 
or,  when  the  axes  are  rectangular, 


Corollary  4.  —  If  we  denote  by  r  the  particular  value 
which  k  assumes  in  the  case  of  a  bisector,  then 

L  +  rL'  ==  0 
represents  the  external,  and 

L  —  rL'  =  0 

the  internal  bisector  of  the  angle  between  the  lines  L=  0 
and  L1  =  0. 


116  ANALYTIC  GEOMETRY. 

In  these  expressions,  r  is  to  be  determined  from  the 

relation 

I/  (A  2  +  B  2  -  2  A  B  cos  co) 

r  =    L  T/(A'2  +  J?'2  —  2  ^!'J3'  cos  w)  : 
which,  in  case  the  axes  are  rectangular,  becomes 


110.  Equation  to  a  right  line  situated  at  in- 
finity. —  To  assume  that   a   right  line  is  at  an  infinite 
distance  from  the  origin  is  to  assume  that  its  intercepts 
on  the  axes  are  infinite.     Hence,  we  have 

c  c 

—  -r  =  GO    and    -  -  -5  —  oo. 

That  is,  supposing  C  to  be  finite, 

A  =  0  and  B  =  0. 
The  required  equation  is  therefore 

Oz  +  Oy  +  0=0: 

in  which  C  is  finite.     We  shall  cite  it  in  the  somewhat 
inaccurate  but  very  convenient  form 

0=0. 

Remark.  —  The  student  will  of  course  remember  that  a  line  at 
infinity  is  not  a  geometric  conception  at  all  —  in  fact  does  not  exist, 
in  any  sense  known  to  pure  geometry.  As  an  analytic  conception, 
however,  it  has  important  bearings;  and  the  equation  just  obtained 
is  useful  in  some  of  the  higher  investigations  of  curves. 

111.  Equations  of  Condition.  —  When  elements  of 
position  and  form  sustain  certain  geometric  relations  to 
each  other,  the  constants  which  enter  their  analytic  equiv- 
alents  must  sustain  corresponding  relations.     In  other 


THREE  POINTS  ON  ONE  EIGHT  LINE.          117 

words,  if  the  geometric  relations  exist,  the  constants  satisfy 
certain  equations.  Such  equations  are  called  Equations 
of  Condition. 

Thus  we  saw  (Art.  97,  Cor.  1)  that  if  two  right  lines 
Ax  +  By  -f  C=  0  and  A'x  +  .B'y  +  C"  =  0  are  parallel, 
the  constants  A,  J9,  Af,  B'  must  satisfy  the  equation 


and  that  if  they  are  mutually  perpendicular,  the  constants 
must  satisfy  the  equation 

AM  +  BB'  —  (AB1  +  A'B)  cos  co  ===  0. 

112.  Condition  that  Three  Points  shall  lie  on 
One  Right  Line.  —  Ifz}y},  x$2,  x^/3  lie  on  one  right  line, 
x$3  must  satisfy  the  equation  to  the  line  which  passes 
through  x}yt  and  x$2.  Hence,  (Art.  95,)  the  equation 
of  condition  is 

(y\  —  2/2)  %z  —  fci  —  x2)  ?/3  +  x}y2  —  ylx2  =  0  : 
which  may  for  the  sake  of  symmetry  be  written 

y,  (x2  —  x.)  -f-  y2  (x.,  —  x})  +  y3  (x,  —  x2)  =  0. 


'/^~~^^ 
Xl(        jj 
N    —  '* 


Remark.  —  It  is  worth  while  to  notice  the  order  of  the  elements 
which  enter  into  the  latter  form  of  this  equation.  In  writing  sym- 
metrical forms,  the  analogous  symbols  must  be  taken  in  a  fixed  order, 
which  will  be  best  understood  by  conceiving  of  the 
successive  symbols  as  forming  a  circuit,  about  which 
we  move  according  to  the  annexed  diagram.  Thus, 
as  in  the  last  equation,  we  pass  from  x\  to  .r2,  from  x2 
to  #3,  and  from  x3  to  x\  ;  and  so  round  again  :  always 
going  back  to  the  FIRST  element  when  the  list  has  been  completed,  and 
then  proceeding  as  before  in  numerical  succession.  The  advantage 
of  symmetrical  forms  is  very  decided,  especially  when  we  have  to 
compare  or  combine  analogous  equations.  But  unless  this  order  is 
observed,  the  methods  of  reasoning  based  upon  it  will  of  course 
fail  ;  and,  in  some  cases,  false  conclusions  may  be  drawn  by  com- 
bining equations  according  to  rules  which  presuppose  it. 


118  ANALYTIC  GEOMETRY. 

113.  Condition  that  Three  Right  Lines  shall 
meet  in  One  Point.  —  If  three  lines  Ax  +  By  -f  C=  0, 

A'x  -f  B'y  +  C'  =  Q,  A"x  -f  £"?/  -f  <7"  =  0  pass  through 
the  same  point,  the  co-ordinates  of  intersection  for  the 
first  two  must  satisfy  the  equation  to  the  third.     Hence, 
(Art.  106,)  the  required  condition  is 
A"  (BC'—Br  C)  +B"  (  CA'—  C'A]  +  C"  (AB'—A'B)  =  0. 

114.  A  condition  often  more  convenient  in  practice, 
is  derived  from   the  principle  of  Arts.  107,  108.     For 
if  three  right  lines  meet  in  one  point,  the  equation  to 
the  third  must  be  in  the  form  of  the  equation  to  a  right 
line   passing   through  the  intersection   of  the  first  and 
second.     Therefore,    supposing    the    three    lines    to    be 
Z  —  0,  L'  =  0,  L"  =  0,  we  can  always  find  some  three 
constants,  Z,  m,  and  —  n,  such  that 

-nL"  =  lL+mL'9 

where  —  n  may  be  either  a  positive  or  a  negative  quan- 
tity.    Hence,  the  condition  is 


That  is,  three  right  lines  meet  in  one  point  when  their 
equations,  upon  being  multiplied  respectively  by  any 
three  constants  and  added,  vanish  identically. 

Remark.  —  For  brevity,  we  shall  often  refer  to  three 
right  lines  passing  through  one  point  by  the  name  of 
convergent*. 

115.  Condition  that  a  Movable  Right  tine  shall 
pass  through  a  Fixed  Point.  —  Comparing  Art.  101, 
Cor.  1  with  Art.  91,  I,  Cor.  1,  it  is  evident  that  the 
equation  to  a  right  line  passing  through  a  fixed  point 
whose  co-ordinates  are  I  :  n  and  m  :  n,  may  be  written 

A  (nx  —  T)  -±  B  (ny  —  m)  =  0. 


MOVABLE  LINE  THROUGH  FIXED  POINT.     119 

And  comparing  this  with  the  general  equation  to  a  right 
line,  we  have 

—  (lA  +  mB)=nC. 

The  required  condition  is  therefore 


Hence,  a  movable  right  line  passes  through  a  fixed  point,  so 
long  as  the  co-efficients  in  its  equation  suffer  no  change  in- 
consistent with  their  vanishing  when  multiplied  each  by  a 
fixed  constant  and  added  together. 

Corollary.  —  The  criterion  of  this  article  may  be  other- 
wise taken  as  the  condition  that  any  number  of  lines 
shall  pass  through  one  point.  For  every  line  whose 
co-efficients  satisfy  the  equation  IA  -f-  mB  -\-  nC  =  0, 
must  pass  through  the  point  I  :  n,  m  :  n. 

116.  In  this  connection  also,  Art.  108  furnishes  us 
with  a  second  condition.  For,  since  we  may  always 
regard  a  fixed  point  as  the  intersection  of  two  given 
right  lines,  the  most  general  expression  for  a  right  line 
passing  through  a  fixed  point  is 

L  +  kL'  -=  0. 

By  writing  L  and  L'  in  full,  and  collecting  the  terms, 
this  becomes 

(A  +  kAr)  x  +  (B+  kBr)  y  +  (C  +  kC'}  =  0. 

Now  the  condition  that  the  line  shall  be  movable  is  that 
k  be  indeterminate.  Hence,  a  movable  ric/ht  line  passes 

c/  J. 

through  a  fixed  point  whenever  its  equation  involves  an 
indeterminate  quantity  in  the  first  degree. 

Corollary.  —  The  co-ordinates  of  the  fixed  point  may 
be  found  by  throwing  the  given  equation  into  the  form 
L  -f  kL'  =  0,  and  solving  L  =  0  and  L'  ==  0  for  x  and  y. 


120  ANALYTIC  GEOMETRY. 

117.  A  third  condition  may  be  obtained  as  follows : 
Suppose  that  we  have  an  equation  of  the  form 

(Axf  +  Byf+C)*  +  (^  #~+*&tf+C')Jl 

+  (A"J  +  B'ty  -f  C")    \  ~~ 

in  which  xf ,  y'  are  the  co-ordinates  of  any  point  on  the  line 
MX'  -{-  Ny'  -f  P=0. 

By  means  of  the  latter  relation,  we  can  eliminate  y'  from 
the  given  equation,  which  will  then  contain  the  indeter- 
minate x'  in  the  first  degree.  Therefore,  a  movable  rigid 
line  passes  through  a  fixed  point  whenever  its  equation 
involves  in  the  first  degree  the  co-ordinates  of  a  point 
which  moves  along  a  given  right  line. 

118.  If  in  any  equation  Ax+  By  -{-  C=Q,  C :  A  or 
C  :  B  is  constant,  the  corresponding  line  (Art.  91,  II, 
Cor.  1)  makes  a  constant  intercept  on  one  of  the  axes. 

Hence,  as  a  fourth  condition,  a  movable  right  line 
passes  through  a  fixed  point  whenever  its  equation  satisfies 
the  relation  C  :  A  —  constant  or  C  :  B  —  constant. 

Corollary. — A  particular  case  of  this  is,  that  the  line 
passes  through  a  fixed  point  when  (7—0.  And,  in  fact, 
(Art.  63)  every  such  line  does  pass  through  the  origin. 

Remark — It  has  seemed  worth  while  to  present  the  condition 
of  this  article  separately,  as  it  is  often  convenient  in  practice.  It 
is  obviously,  however,  only  a  particular  case  of  Art.  116. 

EXAMPLES   ON   THE   RIGHT   LINE. 
I.    NOTATION  AND  CONDITIONS. 

119.  In    some    of   the    exercises    which    follow,    the 
student  must  use  his  judgment  as  to  the  selection  of  the 
axes.     The  labor  of  solving  will  be  much  lessened  by 
a  judicious  choice.     A  few  hints  have  been  given  where 
they  seemed  necessary. 


EXAMPLES  ON  THE  EIGHT  LINE.  121 

1  .  Form  the  equations  to  the  sides  of  a  triangle,  the  co-ordi- 
nates of  whose  vertices  are  (2,  1),  (3,  —  2).  (  —  4,  —  1). 

2.  The  equations   to  the  sides  of  a  triangle   are  x  -{-  y  —  2, 
x  —  3v/  =  4   and    3x  +  5y  -|-  7  =  0  :    find   the    co-ordinates   of  its 
vertices. 

3.  Form  the  equations  to  the  lines  joining  the  vertices  of  the 
triangle  in  Ex.  1  to  the  middle  points  of  the  opposite  sides. 

4.  Form  the  equation  to  the  line  joining  a/?/  to  the  point  mid- 
way between  ,r//y//  and    x///'i////;    or,    show   that,  in   general,  the 
equations  to  the  lines  from  the  vertices  of  a  triangle  to  the  middle 
points  of  the  opposite  sides  are 


(y"  +/"-2/  )x-(x"  +x"'—2x'  )y+(x"  y'  -y"  x'  )+(x'"yr  -y'"x'  )=0, 
(y"'+y'  -2y"  )x-(x"'+x'  -2x"  )y+(x'"y"-y"  fx")+(xt  y"  -y'  x")=Q, 
(/  +y"  -2y"'}x-(x'  +x"  -2x'")y+(x'  y'"-yf  x"'}+(x"y"'-y"y">  )=Q. 

5.  In  the  triangle  of  Ex.  1,  form  the  equations  to  the  perpen- 
diculars from  each  vertex  to  the  opposite  side.     What  inference  as 
to  the  shape  of  the  triangle  ? 

6.  Prove  that,  in  general,  the  equations  to  such  perpendiculars 
are 

(x"  -x"')x  +  (y"  -y"')y  +  (x'   x'"  -\-y'  y'")-(x>   x"  -f  y'  y"  )  =  Q, 

>    +/'  y>    )-(x"  x'"-\-y"  y'"}  =  Q, 
"         '""-x'"x'    --'">      =  0. 


7.  Prove  that  the  general  equations  to  the  perpendiculars  through 
the  middle  points  of  the  sides  are 


2  -*'"2)  -f 

'*-x>   2)  4-  y2(y'"*-y'   2), 

(*'  -x"  )  x  -f  (y  -y"  )y=Y2(x'  2  -x"  2)  -}-  y,  (y>  *  -y"  2). 

8.  Find  the  angle  between  the  lines  x-\-y  =  \  and  y  —  x=  2, 
and  determine  their  point  of  intersection. 

9.  Write  the  equation  to  any  parallel  of  x  cos  a  -f-  y  sin  a  —  p  —  0. 
Decide  whether  x  sin  a  —  ?/  cos  a=p/  or  x  sin  a  -j-  y  cos  a  =p"  may 
be  parallel  to  it;  and,  if  so,  on  what  condition. 


122  ANALYTIC  GEOMETRY. 

10.  Taking  for  axes  the  sides  a  and  b  of  any  triangle,  form  the 
equation  to  the  line  which  cuts  off  the  mih  part  of  each,  and  show 
that  it  is  parallel  to  the  base.     What  condition  follows  from  this? 

11.  Prove  that  y=  constant  is  the  equation  to  any  parallel  of  the 
axis  of  x,  and  x  •=  constant  the  equation  to  any  parallel  of  the  axis 
of  y,  whether  the  axes  are  rectangular  or  oblique. 

12.  Two  lines  AB,  CD  intersect  in  O;  the  lines  AC,  BD  join 
their   extremities  and  meet  in  E  ;   the  lines  AD,  BC  join   their 
extremities  and  meet  in  F:   required  the  condition  that  EF  may 
be  parallel  to  AB. 

13.  Form  the  equation  to  the  line  which  passes  through  (2,  3) 
and  makes  with  the  line  y=3x  an  angle  0  =  60°. 

14.  Form  the  equation  to  the  line  which  passes  through  (2,  —  3) 
and  makes  with  the  line  3x  —  -4y  =  0  an  angle  6  =  —  45°. 

15.  We  have  shown  that,  in  rectangular  axes,  —  A  :  B  =  tan  a, 
but  that,  in  oblique  axes,  —  A  :  B  -—  sin  a  :  sin  (w  —  a).    Prove  that, 
in  all  cases,  tan  a  =  A  sin  «  :  (A  cos  w  —  B). 

16.  Axes  being  oblique,  show  that  two  lines  will  make  with  the 
axis  of  x  angles  equal  but  estimated  in  opposite  directions  (one 
above,  the  other  below)  upon  the  condition 

B      B/ 


1  7.  Find  the  length  of  the  perpendicular  from  (3,  —  4)  on  the 
line  4x  +  2y  —  7  =••  0,  when  <*>  =  60°.  On  which  side  of  the  line 
is  the  given  point? 

18.  Find  the  length  of  the  perpendicular  from  the  origin  on  the 
line  a  (x  —  a)  +  b  (y  —  b)  =  0. 

19.  Given  the  equations  to  two  parallel  lines:  to  find  the  distance 
between  them. 

20.  What  points  on  the  axis  of  x  are  at  the  distance  a  from  the 


21.  Form  the  equation  to  the  bisectors  of  the  angles  between 
Zx  -f-  4y  —  9=0  and  \"2x  -j-  5y  —  3  =  0  ;  -  the  equation  to  any 
right  line  passing  through  their  intersection. 


EXAMPLES  ON  THE  RIGHT  LINE.  123 

22.  Prove  that  whether  the  axes  be  rectangular  or  oblique  the 
lines  x-\-  y  =-Q,  x  —  y  =  0  are  at  right  angles  to  each  other,  and 
bisect  the  supplemental  angles  between  the  axes.    Show  analytically 
that  all  bisectors  of  supplemental  angles  are  mutually  perpendicular. 

23.  Find  the  equation  to  the  line  passing  through  the  intersec- 
tion of  3x —  5y  -\-  6  =  0  and  2x  -f  y  —  8  =  0,  and  striking  the  point 
(5,  6). 

24.  Find  the  equation  to  the  line  joining  the  origin  to  the  inter- 
section of  Ax  +  By  +  C^--  0  and  A'x  +  B'y  +  C'  =  0. 

25.  Show  that  the  equation  to  the  line  passing  through  the  inter- 
section of  Ax  +  By  +  C=  0  and  A'x  +  B'y  -f  C=  0,  and  parallel 
to  the  axis  of  *,  is  (A  B'  —  A'B)  y+  (A  C/ —  A'C)  =  0.    Does  this 
agree  with  the  theorem  of  Ex.  11? 

26.  Find  the  equation  to  the  line  passing  through  the  intersec- 
tion of 

x  cos  a  -f-  ysina  =p,  x  cos  ft  -f-  y  sin  /3  =p' 

and  cutting  at  right  angles  the  line 

x  cos  7  +  y  sin  y  =p". 

27.  Given  any  three  parallel  right  lines  of  different  lengths ;  join 
the  adjacent  extremities  of  the  first  and  second,  and  produce  the 
two  lines  thus  formed  until  they  meet;  do  the  same  with  respect 
to  the  second  and  third,  and  the  third  and  first:  the  three  points 
of  intersection  lie  on  one  right  line. 

28.  Given  the  frustum  of  a  triangle,  with  parallel  bases:   the 
intersection  of  its  diagonals,  the  middle  points  of  the  bases,  and 
the  vertex  of  the   triangle   are  on  one  right  line.     [Take  vertex 
of  triangle  as  origin,  and  sides  for  axes.] 

29.  On  the  sides  of  a  right  triangle  squares  are  constructed ; 
from  the  acute  angles  diagonals  are  drawn,  crossing  the  triangle  to 
the  vertices  of  these  squares ;  and  from  the  right  angle  a  perpen- 
dicular is  let  fall  upon  the  hypotenuse :  to  prove  that  the  diagonals 
and  the  perpendicular  meet  in  one  point.     [Let  the  lengths  of  the 
sides  be  a  and  6,  and  take  them  for  axes] 

30.  Prove  that  the  three  lines  which  join  the  vertices  of  a  tri- 
angle to  the  middle  points  of  the  opposite  sides  are  convergents, 
taking  for  axes  any  two  sides.     [See   also  their  equations  above, 
Ex.  4.] 


124  ANALYTIC   GEOMETRY. 

31.  The  three  perpendiculars  from  the  vertices  on  the  sides,  and 
the  three  that  rise  from  the  middle  points  of  the  sides,  are  each 
convergents.     [See  their  equations,  Exs.  6  and  7.] 

32.  The  three  bisectors  of  the  angles  in  any  triangle  are  con- 
vergen^s:   for  their  equations  are 

(x  cos  a  -f-  y  sin  a  —  p  )  —  (x  cos  ft  +  y  sin  /?  —  p/  )  =  0, 
(x  cos  j3  +  y  sin  (3  —  p'  )  —  (x  cos  y  +  y  sin  y  —  p")  =  0, 
(x  cos  7  -f  y  sin  y  —  _p")  —  (x  cos  a  -f-  y  sin  a  —  p  )  =  0, 

if  we  suppose  the  origin  to  be  within  the  triangle. 

33.  Through  what  point  do  all  the  lines  y  =  mx,  Ax  +  By  —  0, 
2x  =  3y,  x  cos  a  +  y  sin  a  =  0,  p  cos  0  =  0  pass  ? 

34.  Decide  whether  the  lines  2x  —  3y  -f  6  =  0,  4x  -f  3y  —  6  =  0, 
5.r  —  52/+10  =  0,  7x  +  2y  — 4  =  0,  *  —  y-f  2  =  0  pass  through  a 
fixed  point. 

35.  Given  the  line  3x  —  5y-\-&=Q:   form  the  equations  to  five 
lines  passing  through  a  fixed  point,  and  determine  the  point. 

36.  Given  three  constants  2,  3,  5 :  form  the  equations  to  five  lines 
passing  through  a  fixed  point,  and  determine  the  point. 

37.  Given  the  vertical  angle  of  a  triangle,  and  the  sum  or  differ- 
ence of  the  reciprocals  of  its  sides:  the  base  will  move  about  a  fixed 
point. 

38.  If  a  line  be  such  that  the  sum  of  the  perpendiculars,  each 
multiplied  by  a  constant,  let  fall  upon  it  from  n  fixed  points,  is  =  0,  it 
will  pass  through  a  fixed  point  known  as  the  Center  of  Mean  Position 
to  the  given  points. 

The  conditions  of  the  problem  (Art.  105,  Cor.  1)  give  us 

m'  (x'  cos  a  +  y'  sin  a  —p)  +  m"    (x"   cos  a  +  y"   sin  a  —  p)  | 

-f  m"1  (x'"  cos  a  +  y'"  sin  a  — p)  -f  <fcc.  j  ~     > 

or,  putting  Z(mx')as  an  arbitrary  abbreviation  for  the  sum  of  the  mx's, 
2  (TO?/')  for  the  sum  of  the  my's,  and  £(»i)  for  the  sum  of  the  m's, 

X  (mx' )  cos  a  +  2  (my' )  sin  a  —  2  (TO)  p  =  0. 

Solving  for  p,  and  substituting  its  value  in  xcos  a  +  y  sin  a  —  p  =  0,  the 
equation  to  the  movable  line  becomes 

2  (m)  x—Z  (mx1 )  +  tan  o  {  2  (TO)  y  -  2  (my1 ) }  =  0  : 
which  (Art.  116)  proves  the  proposition.     [Solution  by  Salmon.] 


RECTILINEAR  LOCI.  125 

39.  If  the  three  vertices  of  a  triangle  move  each  on  one  of  three 
convergents,  and  two  of  its  sides  pass  through  fixed  points  x'y',  xf'y", 
the  third  will  also  pass  through  a  fixed  point.    [Take  the  two  exterior 
convergents  for  axes.] 

40.  If  the  vertex  in  which  the  two  sides  mentioned  in  Ex.  39 
meet,  does  not  move  on  a  line  convergent  with  the  two  on  which 
the  other  vertices  move,  to  find  the  condition  that  the  third  side 
may  still  pass  through  a  fixed  point. 

II.    RECTILINEAR   LOCI. 

12O.  The  following  examples  illustrate  the  method 
of  solving  problems  in  which  the  path  of  a  moving  point 
is  sought.  When  a  point  moves  under  given  conditions, 
we  have  only  to  discover  what  relation  between  its  co-ordinates 
is  implied  in  those  conditions :  then  by  writing  this  relation 
in  algebraic  symbols  we  at  once  obtain  the  equation  to  its 
locus. 

As  the  investigation  of  loci  is  one  of  the  principal  uses 
of  Analytic  Geometry,  it  is  important  that  the  student 
should  early  acquire  skill  in  thus  writing  down  any 
geometric  condition.  The  relation  between  the  co-ordi- 
nates is  sometimes  so  patent  as  to  require  no  investiga- 
tion :  for  instance,  if  we  were  required  to  determine  the 
locus  of  a  point  the  sum  of  the  squares  on  whose  co-ordi- 
nates is  constant,  we  should  at  once  write  the  equation 

%2  -f-  y2  —  r2 

and  discover  (Art.  25,  II)  that  the  locus  is  a  circle. 
But  as  a  general  thing  the  relation  must  be  developed 
from  the  conditions  by  means  of  the  geometric  or  trigo- 
nometric properties  which  they  imply.  The  process  may 
be  much  simplified  by  a  proper  choice  of  axes.  As  a  rule, 
the  equations  are  rendered  much  shorter  and  easier  to 
interpret  by  taking  for  axes  two  prominent  lines  of  the 
figure  to  which  the  conditions  give  rise.  Still,  by  taking 


126 


ANALYTIC   GEOMETRY. 


axes  distinct  from  the  figure,  we  sometimes  obtain  equa- 
tions whose  symmetry  more  than  compensates  for  their 
loss  of  simplicity.  For  instance,  in  the  equations  of  Exs. 
4,  6,  and  7,  when  the  first  is  obtained,  the  second  and 
third  can  be  written  out  at  once  by  analogy. 

1.  Given  the  base  of  a  triangle  and  the  difference  between  the 
squares  on  its  sides  :  to  find  the  locus  of  its  vertex. 

Let  us  take  for  axes  the  base  and  a  perpendicular  at  its  middle  point. 
Let  the  base  ==  2m,  and  the  difference  between  the 
squares  on  the  sides  =  ri1.  Then,  the  co-ordinates 
of  the  vertex  being  x  and  y,  we  shall  have  RP~  = 
r/2  +  (m  +  x)2  and  PQ1  =  f  +  (m  —  x)2.  That  is, 
?/2  +  (m  +  x)2  —  \yz  +  (m  —  x)2}  =  n2  ;  or  the  equa- 
tion to  the  required  locus  is 

4  mx  =  n2. 

Hence  (Art.  25,  I)  the  vertex  moves  upon  a  right  line  perpendicular  to  the 
base. 

2.  Find  the  locus  of  the  vertex  of  a  triangle,  given  the  base  and 
the  sum  of  the  cotangents  of  the  base  angles. 

Using  the  same  axes  as  in  Ex.  1,  and  putting  cot  R  +  cot  Q  —  n,  we 
have  from  the  diagram  cot  R  —  —  —  and  cot  Q  =  —  —  .  Hence,  the 
equation  to  the  required  locus  is 


and  (Art.  25,  I)  the  vertex  moves  along  a  line  parallel  to  the  base  at  a 
distance  from  it  =  2m  :  n. 

3.  Given  the  base  and  that  rcot-R=fc  scotQ=p  ±  g,  find  the 
locus  of  the  vertex. 

4.  Given  the  base  and  the  sum  of  the  sides,  let  the  perpendicular 
to  the  base  through  the  vertex  be  produced  upward  until  its  length 
equals  that  of  one  side :  to  find  the  locus  of  its  extremity. 

5.  Given  two  fixed  lines  OM,  ON;  if  any 
line  MN  parallel  to  a  third  fixed  line  OL  in- 
tersect them :   to  find  the  locus  of  P  where 
MN  is  cut  in  a  given  ratio,  so  that  MP  = 
nMN.  CV 


RECTILINEAR  LOCI.  127 

Here  it  will  be  most  convenient  to  take  for  axes  the  two  fixed  lines 
OM,  OL.  Then,  since  N  is  &  point  on  the  line  ON,  we  shall  have 
MN=mOM;  and,  substituting  in  the  given  equation,  we  find  the  equa- 
tion to  the  required  locus,  namely, 

y  =  mnx. 

Hence  (Art.  78,  Cor.  5)  the  point  of  proportional  section  moves  on  a  right 
line  passing  through  the  intersection  of  the  two  given  lines. 

6.  Find  the  locus  of  a  point  P,  the  sum  of  whose  distances  from 
O3/,  OL  is  constant. 

7.  A  series  of  triangles  whose  bases  are  given  in  magnitude  and 
position,  and  whose  areas  have  a  constant  sum,  have  a  common 
vertex  :  to  find  its  locus. 

8.  Given  the  vertical  angle  of  a  triangle,  and 
the  sum  of  its  sides :    to  find  the  locus  of  the 
point  P  where  the  base  is  cut  in  a  given  ratio, 

so  that  mPX  =nPY.  TJ     M  x 

9.  Determine  the  locus  of  P  in  the  annexed  diagram  under  the 
following    successive    conditions;     PQ    being 
perpendicular  to  OQ,  and  PR  to  OR: 

i.  OQ  +  OR  =  constant.  N 

n.  QR  parallel  to  y  =mx. 


in.  QR  cut  in  a  given  ratio  by  y=mx  +  l>.       °          M    Q 

121.  Hitherto  we  have  found  the  equations  to  required 
loci  by  expressing  the  given  conditions  directly  in  terms 
of  the  variable  co-ordinates.  But  it  is  often  more  con- 
venient to  express  them  at  first  in  terms  of  other  lines,  and 
then  find  some  relation  between  these  auxiliaries  by  which 
to  eliminate  the  latter:  the  result  of  eliminating  will  be 
an  equation  between  the  co-ordinates  of  the  point  whose 
locus  is  sought. 

This  process  of  forming  an  equation  by  eliminating  an 
indeterminate  auxiliary   is   extensively   used,  and    is    of 
especial  advantage  when  investigating  the  intersections 
of  movable  lines. 
An.  Ge.  14. 


128  ANALYTIC  GEOMETRY. 

10.  Given  the  fixed  point  A  on  the  axis  of  re,  and  the  fixed  point 
B  on  the  axis  of  y ;  on  the  axis  of  x  take  any  point  A',  and  on  the 
axis  of  y  any  point  B',  such  that  OA/  -f  OB'  =  OA  +  O£ :  to  find 
the  locus  of  the  intersection  of  AB/  and  A/B. 

If  OA  =  a,  and  0#  =  6  .  • .  OA'  =  a  +  (k  indeterminate),  and 
OB'  =  b  —  k.  Hence,  (Art.  79,)  by  clearing  of  fractions  and  collecting 
the  terms,  the  equations  to  AB'  and  A' B  may  be  written 

bx  +  ay  —  ab  +  k  (a  —  x)  =  0, 
bx  +  ay  —  ab  +  k  (y  —  b)  =  0. 

Subtracting,  we  eliminate  the  indeterminate  k;    and  the  equation  to  the 
required  locus  is 

x  +  y  —  a  +  b.  # 

11.  In  a  given  triangle,  to  find  the  locus  of  the  middle  point  of  the 
inscribed  rectangle. 

12.  In  a  given  parallelogram,  whose  adjacent  sides  are  a  and  6, 
to  find  the  locus  of  the  intersec- 
tion of  AB'  and  A'B :  the  lines 

A  A',  BB/  being  any  two  paral-  y A'^- 

lels  to  the  respective  sides.  /    „---  ""7     /,>• 

[The  statement  of  this  problem  /                 /Sf 

will  apparently  involve  two  indeter-      / /-•''   / 

minate  quantities  j  but  both  can  be 
eliminated  at  one  operation.] 

13.  A  line  is  drawn  parallel  to  the  base  of  a  triangle,  and  the 
points  where  it  meets  the  two  sides  are  joined  transversely  to  the 
extremities  of  the  base :  to  find  the  locus  of  the  intersection  of  the 
joining  lines. 

14.  Through  any  point  in  the  base  of  a  triangle  is  drawn  a  line 
of  given  length  in  a  given  direction :  supposing  it  to  be  cut  by  the 
base  in  a  given  ratio,  find  the  locus  of  the  intersection  of  the  lines 
joining  its  extremities  to  those  of  the  base. 

15.  Given  a  point  and  two  fixed  right  lines;  through  the  point 
draw  any  two  right  lines,  and  join  transversely  the  points  where 
they  meet  the  fixed  ones :  to  find  the  locus  of  the  intersection  of  the 
joining  lines. 


*See  Salmon's  Conic  Sections,  p.  46. 


PAIRS  OF  RIGHT  LINES. 


129 


When  we  have  to  determine  the  locus  of  the 
extremity  of  a  line  drawn  through  a  fixed  point  under 
given  conditions,  it  is  generally  convenient  to  employ 
polar  co-ordinates.  We  make  the  fixed  point  the  pole, 
and  then  the  distance  from  it  to  the  extremity  of  the 
moving  line  becomes  the  radius  vector. 

16.  Through  a  fixed  point  O  is  drawn  a  line  OP,  perpendicular 
to   a   line  QR  which  passes  through  the  fixed 
point  Q  :  to  find  the  locus  of  its  extremity  P, 
if  OP.  OR  =  constant. 

Let  the  distance  OQ  =  a,  and  let  OP.OR=m*. 
From  the  diagram,  OR  =  OQ  cos  ROQ.  Hence,  the 
equation  to  the  locus  of  P  is 

pa  cos  0  =  mz ; 


Q 


and  (Art.  82,  Cor.)  P  moves  on  a  right  line  perpendicular  to  OQ. 

17.  One  vertex  of  an  equilateral  triangle  is  fixed,  and  the  second 
moves  along  a  fixed  right  line :  to  find  the  locus  of  the  third. 

18.  In  a  right  triangle  whose  two  sides  are  in  a  constant  ratio, 
one  acute  angle  has  a  fixed  vertex,  and  the  vertex  of  the  right  angle 
moves  on  a  given  right  line:    to  find  the  locus  of  the  remaining 
vertex. 

19.  Given  the  angles  of  any  triangle :  if  one  vertex  is  fixed,  and 
the  second  moves  on  a  given  right  line,  to  find  the  locus  of  the  third. 

[The  student  will  readily  perceive  that  Exs.  17  and  18  are  particular 
cases  of  Ex.  19.  He  should  trace  this  relation  through  the  equations  to 
the  corresponding  loci.] 

20.  Given   the   base  of  a  triangle,   and  the  sum  of  the  sides; 
through  either  extremity  of  the  base,  a  perpendicular  to  the  adjacent 
side  is  drawn :  to  find  the  locus  of  its  intersection  with  the  bisector 
of  the  vertical  angle. 


SECTION   III.  —  PAIRS   OF   RIGHT   LINES. 


Since  the  equation  of  the  first  degree  always 
represents  a  right  line,  there  is  but  one  locus  whose  equa- 
tion is  of  the  first  degree.  But  there  are  several  loci 


130  ANALYTIC  GEOMETRY. 

whose  equations  are  of  the  second  degree  ;  and,  in  accord- 
ance with  the  principle  of  Art.  76,  we  shall  consider  these 
separately  before  discussing  their  relations  to  the  locus 
of  the  Second  order  in  general.  In  the  case  of  each, 
we  shall  first  obtain  its  equation,  and  then  find  the  con- 
dition on  which  the  general  equation  of  the  second  degree 
will  represent  it.  When  this  has  been  done  for  all  of 
them,  we  can  pass  to  the  purely  analytic  ground,  and 
show  that  the  equation  of  the  second  degree  always 
represents  one  of  these  lines. 

We  begin  with  the  cases  in  which  it  represents  a  pair 
of  right  lines.  These  have  a  special  interest,  as  furnish- 
ing the  title  by  which  the  Right  Line  takes  its  place  in 
the  order  of  Conies. 

I.    GEOMETRIC  POINT  OF  VIEW  :  —  THE  EQUATION  TO  A  PAIR 
OF  RIGHT  LINES  IS  OF  THE  SECOND  DEGREE. 

124.  Any  equation  in  the  type  of  LMN.  .  .  =  0  will 
obviously  be  satisfied  by  supposing  either  of  the  factors 
L,  Mj  N)  etc.,  equal  to  zero.     If,  then,  L=  0,  M=  0, 
N=  0,  etc.,  are  the  equations  to  n  different  lines,  their 
product  will  be  satisfied  by  any  values  of  x  and  y  that 
satisfy  either  of  them.     That  is,  LMN.  .  .  =  0  will  be 
satisfied  by  the  co-ordinates  of  any  point  on  either  of  the 
n  lines:  its  locus  is  therefore  the  group  of  lines  severally 
denoted  by  the  separate  factors.     Hence,  we  form  the 
equation  to  a  group  of  lines  by  multiplying  together  the 
equations  to  its  constituents. 

125.  Equation   to   a   Pair   of  Right   Lines.  —  By 
the  principle  just  established,  the  required  equation  is 


or,  by  writing  the  abbreviations  in  full,  expanding,  and 
collecting  the  terms, 


PAIR  OF  LINES  THRO  UGII  FIXED  POINT.       131 

X+&QV 


+A"C' 

which  is  manifestly  a  particular  case  of  the  general  equa- 
tion of  the  second  degree  in  two  variables, 

Ax2  -f  2Hxy  +  By2  -\-ZGx-\-  %Fy  -f  C  —  0,  * 
in  which  J.,  5,  (7,   F,  G,  H  are  any  six  constants. 

126.  Equation  to  a  Pair  of  Right  Lines  passing 
through  a  Fixed  Point. — The  equations  to  two  right 
lines  passing  through  the  point  x'y'  (Art.  101,  Cor.  1) 
are 

A'(x  -  d)+B'  (y  -  y1}  =  0,  A»(x  -  x')-tB"(y-y')  ==  0. 
Hence,  (Art.  124,)  the  required  equation  is 


or,   since   A'  A",   A'B"+A"B',   B'B"  are  independent 
of  each  other, 


A(x  -  ^  +  2H(x  -  aO(y  -  y')  +  B(y  -  yj  =  0. 

Corollary  1,  —  The  equation  to  a  pair  of  right  lines 
passing  through  the  origin  (Art.  49,  Cor.  3)  will  be 

Ax2  +  2Hxy  +  By2  =  0. 

Corollary  2.  —  Of  the  two  equations  last  obtained,  the 
former  is  evidently  homogeneous  with  respect  to  (x  —  xf) 
and  (?/  —  y'}  ;  and  the  latter,  with  respect  to  x  and  y. 
Hence,  Every  homogeneous  quadratic  in  (x  —  x')  and 


'*"  The  equation  of  the  second  degree  in  two  variables  is  usually  written 

Ax2  +  Bxy  +  Cy  +  Dx  +  EI/  +  F^  0. 

I  have  followed  Salmon  in  departing  from  this  familiar  expression.  The 
sequel  will  show,  however,  that  the  new  form  imparts  to  the  equations 
derived  from  it  a  simplicity  and  symmetry  which  far  outweigh  any  incon- 
venience that  may  arise  from  its  unfamiliar  appearance. 


132  ANALYTIC  GEOMETRY. 

(j  ~JO  represents  two  right  lines  passing  through  the  fixed 
point  x'y'  ;  and,  Every  homogeneous  quadratic  in  x  and  j 
represents  two  right  lines  passing  through  the  origin. 

127.  The  equation  Ax2  -f-  2Hxy  -f-  By2  —  0  deserves 
a  fuller  interpretation,  as  it  leads  to  conditions  which 
have  a  most  important  bearing  on  the  relations  of  the 
Right  Line  to  the  Conies. 

If  we  divide  it  by  x2,  it  assumes  the  form  of  a  complete 
quadratic  in  y  :  x, 


But  (Arts.  124  ;  78,  Cor.  5)  it  may  also  be  written 


Hence,  (Alg.,  234,  Prop.  2d,)  its  roots  are  the  tangents  of 
the  angles  made  with  the  axis  of  x  by  the  two  lines  which 
it  represents.  Now  if  we  solve  (1)  for  y  :  x,  we  obtain 

y__       H±V(H2-AB)  , 

x~  B 

that  is,  the  roots  are  real  and  unequal  when  H2  >  AB  ; 
real  and  equal  when  H  2  =  AB  ;  and  imaginary  when 
H2<AB.  Therefore,  if  H2  —  ^45  >0,  the  equation 
denotes  two  real  right  lines  passing  through  the  origin  ; 
if  H2  —  AB  =  0,  two  coincident  ones  ;  but  in  case 
H2  —  AB  <  0,  two  imaginary  ones. 

Corollary.  —  The  reasoning  here  employed  is  obviously 
applicable  to  the  equation  of  Art.  126.  The  meaning 
of  any  homogeneous  quadratic  may  therefore  be  deter- 
mined according  to  the  following  table  of  corresponding 
analytic  and  geometric  conditions  : 


ANGLE  OF  A  PAIR  OF  LINES. 


133 


ff2  -  -  AB  >  0  .  •  .  Two  real  right  lines  passing  through 
a  fixed  point. 

H2  —  AB  —  0  .  •  .  Two  coincident  right  lines  passing 
through  a  fixed  point. 

H2  —  JJ?<0  .-.  Two  imaginary  right  lines  passing 
through  a  fixed  point. 

Note  —  By  thus  admitting  the  conception  of  coincident  and  im- 
aginary lines  as  well  as  of  real  ones,  we  are  enabled  to  assert  that 
every  quadratic  which  satisfies  certain  conditions  represents  two 
right  lines.  In  fact,  the  result  just  obtained  permits  us  to  say  that 
every  equation  between  plane  co-ordinates  denotes  a  line  (or  lines), 
and  to  include  in  this  statement  such  apparently  exceptional  equa- 
tions as 


Of  this  equation,  we  saw  (Art.  61,  Rem.)  that  the  only  geometric 
locus  is  the  fixed  point  (a,  0).  But  it  is  evidently  homogeneous  in 
(x  —  a\  (y  —  0)  and  fulfills  the  condition  H'1  —  AB  <Q.  We 
may  therefore  with  greater  analytic  accuracy  say  that  it  denotes 
two  imaginary  right  lines  passing  through  the  point  (a,  0)  ;  or,  as  we 
shall  see  hereafter,  two  imaginary  right  lines  whose  intersection  is  the 
center  (a,  0)  of  an  infinitely  small  circle. 

Such  statements  may  seem  to  be  mere  fictions  of  terminology; 
but  the  farther  we  advance  into  our  subject,  the  greater  will  appear 
the  advantage  of  thus  making  the  language  of  geometry  correspond 
exactly  to  that  of  algebra.  If  we  neglect  to  do  so,  we  shall  overlook 
many  remarkable  analogies  among  the  various  loci  which  we  in- 
vestigate. 

128.  Angle  of  a  Pair  of  Right  lines.  —  From 
Art.  96  we  have,  as  the  expression  for  determining  this, 


tan^>  = 


mm 


But  we  saw  (Art.  127)  that  m  and  m!  are  the  roots  of  the 
equation  Ax2  +  2  Hxy  +  By2  =  0.     Hence, 


m  —  m  = 


mm  = 


B 


134  ANALYTIC  GEOMETRY. 


Therefore  tan  <p  —  --  v,        p 

-A  ~~f~  JD 

Corollary.  —  The  condition  that  a  pair  of  rigid  lines  shall 
cut  each  other  at  riqht  anqles  is 

t/  t7 


Kemark  —  The  student  can  readily  convince  himself  that  when 
the  axes  are  oblique 

=  2smul/(H2  —  AB) 
~  A  +  B-2Hcosu  ' 

129.  Equation  to  the  Bisectors  of  the  angles 
between  the  Pair  Ax2  +  2  Hay  +  By^=O.  —  The  equa- 
tion (compare  Arts.  124;  109,  Cor  1)  will  be 

(A'x  +  B'y)2  (A"2  +  B"2)  —  (A"x  +  B"y)2  (A*  +  B'2}  =  0. 

Expanding,  collecting  the  terms,  and  dividing  through  by 

A'B"  —  A"B',  this  becomes 

(^/JB//+^//J5/)  x2-  2  (A'  A"-  B'B"}  xy  —  (A'B"+  A"B')y*=  0. 

Comparing  the  co-efficients  with  those  in  the  original 
form  of  the  equation  to  the  two  given  lines  (Art.  126) 
we  obtain 


Corollary  1,  —  The  co-efficients  of  this  equation  satisfy 
the  condition  (Art.  128,  Cor.)  A  +  B  —  0,  and  show  that 
the  two  bisectors  are  at  right  angles  :  which  agrees  with 
the  result  of  Ex.  22,  p.  123. 

Corollary  2.  —  The  equation  to  the  bisectors  is  evidently 
a  quadratic  in  y  :  x  of  the  first  or  second  form  :  its  roots 
are  therefore  always  real,  whether  those  of  the  equation 
to  the  given  pair  are  real  or  not.  Hence  wre  have  the 
singular  result,  that  a  pair  of  imaginary  lines  may  have 
a  real  pair  bisecting  the  angles  between  them.  And  it  is  a 
noticeable  inference  from  the  discussion  in  Art.  127,  that 
two  imaginary  lines  may  have  a  real  point  of  intersection. 


ANALYTIC  CONDITION  FOE  PAIR  OF  LINES.    135 

Remark — These  two  equations,  Ax2  +  "2Hxy  +  By1  =  0  and 
Hx2—  (A  —  B}xy  —  Hif  =  0,  merit  the  student's  special  attention. 
They  will  re-appear,  in  a  somewhat  unexpected  quarter. 

II.    ANALYTIC    POINT    OF    VIEW:  —  THE    EQUATION    OF    THE 

SECOND  DEGREE  ON  A  DETERMINATE  CONDITION 

REPRESENTS  TWO  RIGHT  LINES. 

ISO.  This  theorem  is  an  immediate  consequence  of 
the  method  by  which  we  form  the  equation  to  a  pair  of 
right  lines.  For,  as  that  equation  always  originates  in 
the  expression 

£27=6, 

so,  conversely,  it  must  always  be  reducible  to  this  form. 
Hence,  An  equation  of  the  second  degree  will  represent  two 
right  lines  whenever  it  can  be  resolved  into  two  factors  of 
the  first  degree. 

Corollary. — The  same  reasoning  manifestly  applies  to 
LMN. . .  =  0.  Hence,  an  equation  of  any  degree,  which 
can  be  separated  into  factors  of  lower  degrees,  represents 
the  assemblage  of  lines  separately  denoted  by  the  several 
factors.  In  particular,  if  an  equation  of  the  nth  degree 
is  separable  into  n  factors  of  the  first  degree,  it  represents 
n  right  lines. 

133.  We  may  express  the  condition  just  determined, 
in  the  form  of  a  constant  relation  among  the  co-efficients 
in  the  general  equation  of  the  second  degree. 

The  equation  to  any  pair  of  right  lines  may  be  written 

{y  - (m'x  +  b') }  {y -  (m"x  +  b") }  =  0. 

Hence,  (Alg.,  234,  Prop.  2d,)  in  order  that  the  equation 
of  the  second  degree  may  represent  two  right  lines,  its 
roots  must  assume  the  form  y  —  mx  -f-  b.  Now  if  we 
solve  Ax2  +  2ffxy  -f  By2  -f  2Gx  -f  2Fy  -f  0=  0  as  a 
complete  quadratic  in  ?/,  we  obtain 
An.  C.v  I:.. 


136  ANALYTIC  GEOMETRY. 


and  in  order  that  this  may  assume  the  form  y  =  mx  -J-  b, 
the  expression  under  the  radical  must  be  a  perfect  square. 
But  the  condition  for  this  (or,  in  other  words,  the  condi- 
tion that  the  general  equation  of  the  second  degree  may 
represent  two  right  lines)  is 

(H2—AB)  (F2  —  BO)  =  (HF—BG)2        (1). 

Expanding  and  reducing,  we  may  write  this  in  the  striking 
symmetrical  form 

ABC-\-2FGff—AF2  —  BG2-CR2  =  Q      (2). 

Corollary.  —  It  is  evident  from  (1),  that  H2  —  AB  and 
F2  —  BO  will  be  positive  together  or  negative  together, 
but  will  not  necessarily  vanish  together. 

132.  In  the  light  of  these  results,  it  will  be  interesting 
to  test  the  conditions  H2—  AB>0,  H2—AB  =  Q, 
H2  —  AB  <  0  in  the  general  equation  to  a  pair  of  right 
lines. 

From  (1),  it  follows  that  the  roots  of  this  equation  may 
be  written 


hence  (taking  account  of  the  preceding  corollary)  they 
will  be  real  and  unequal,  when  H2  >  AB;  will  differ  by 
a  constant,  when  H2  =  AB  ;  and  will  be  imaginary,  when 
H2  <  AB.  But  these  roots  are  the  ordinates  of  the  two 
lines  represented  by  the  equation  of  Art.  125  :  hence,  in 
any  equation  of  the  second  degree  whose  co-efficients 
satisfy  the  condition  (2),  we  shall  have  the  following 
criteria  : 

H2  —  AB  >  0  /.  Two  real,  intersecting  right  lines. 

H2—AB  =  0  .'.  Two  parallel  right  lines. 

H2  —  AB  <  0  /.  Two  imaginary,  intersecting  right  lines. 


EXAMPLES  ON  PAIRS  OF  LINES.  137 


Corollary.—  If  H2  —  AB  =  0,  and  F2  —  BC=Q  at 
the  same  time,  the  two  lines  are  coincident.  Hence,  A 
right  line  fanned  by  the  coincidence  of  two  others  is  the 
limit  of  two  parallels. 

EXAMPLES. 

1.  Form  the  equation  to  the  two  lines  passing  through  (2,  3), 
(4,  5)  and  (1,  6),  (2,  5). 

2.  What  locus  is  represented  by  xy  =  0  ?     By  x2  —  y*  =  0  ? 
By  x1  —  5xy  +  6?/2  =  0  ?     By  ;e2  —  2xy  tan  0  —  ya  =  0? 

3.  What  lines  are  represented  by  a?  —  6#2y  -j-  11  xy2  —  6^  =  0? 

4.  Show  what  loci  are  represented  by  the  equations  #2  -f-  y*  =  0, 

x*  +  xy  =  0,  x2  +  3/2  +  «2  =  0,  *y  -  a*  =  0. 

5.  Interpret    (*  —  a)  (?/  —  6)  =  0,    (x  —  a)2+  (y  —  £)2  =  0,   and 
Cr-y+a)2-fO  +  y-a)2:=0. 

6.  Show    that   (y  —  3#  +  3)  (3y  +  *  —  9)  =  0    represents    two 
right  lines  cutting  each  other  at  right  angles. 

7.  Find  the  angles  between  the  lines  in  Ex.  2. 

8.  What  is  the  angle  between  the  lines  x*  -f-  xy  —  67/2=0? 

9.  Write  the  equation  to  the  bisectors  of  the  angles  between 
the  pair  x2  —  5xy  +  6y2  =  0. 

10.  Write  the  equations  to  the  bisectors  of  the  angles  between 
the  pairs  x1  —  y2  =  0  and  x1  —  a?y  +  y2  =  0. 

11.  Show  that  the  pair  6.r2  +  bxy  —  6?/2  =  0  intersect  at   right 
angles. 

12.  Show  that  6ar2  +  5xy  —  6?/2  =  0  bisect  the  angles  between  the 
pair  2x*  +  I2xy  +  ly1  —  6. 

13.  Verify  that  a;2  —  5xy  +  4y2  +  x  -f  2y  —  2  =  0  represents  two 
right  lines,  and  find  the  lines. 

14.  Show  that  9z2  —  I2xy  +  4yz  —  2x  +  y  —  3  =  0  does  not  rep- 
resent right  lines,  and  find  what  value  must  be  assigned  to  the  co- 
efficient of  x  in  order  that  it  may. 

15.  Show  that  4z2  —  I1xy+  9y2  —  4x  +  6y  —  12  =  0  represents 
two  parallel  right  lines,  and  find  the  lines. 


138  ANALYTIC  GEOMETRY. 

16.  Show  that  4x*  —  I2xy  +  9/  —  4x  +  6y  +  1  =  0  denotes  two 
coincident  right  lines,  and  find  the  line  which  is  their  limit. 

17.  Show  that  5z2  —  \1xy  +  9?/2  —  2x  +  &y  +  10  =  0  represents 
two  imaginary  right  lines,  and  find  them. 

18.  Show  that  if  A,  B,  (7,  L,  N  are  any  five  constants,  the  equation 
to  any  pair  of  right  lines  may  be  written 

(Ax  +  By+  C)*--=  (Lx  +  JV)«; 


that  the  lines  are  real,  when  (Lx  +  N)2  is  positive;  imaginary, 
when  (Lx  +  Ny  is  negative;  parallel,  when  i  =  0;  and  coinci- 
dent, when  L  and  N  are  both  =  0. 

19.  Form  an  equation  in  the  type  of  Ax*  +  2Hxy  +  By2  +  2Gx 
+  2J?V+C=0  with  numerical  co-efficients,  which  shall  represent 
two  real  right  lines. 

20.  Form  a  similar  equation  representing  two  parallel  lines,  and 
one  representing  two  coincident  ones;    also,  one  representing  two 
imaginary  lines. 


SECTION  IV. — THE  CIRCLE. 

I.    GEOMETRIC    POINT    OF   VIEW :  —  THE    EQUATION   TO   THE 
CIRCLE   IS    OF    THE    SECOND    DEGREE. 

133.  The  Circle   is   distinguished    by   the   following 
remarkable  property :    The  variable  point  of  the  curve  is 
at  a  constant  distance  from  a  fixed  point,  called  the  center. 

134.  Rectangular  equation   to  the  Circle. — If 

x y  represent  any  point  on  the  curve,  and  r  its  constant 
distance  from  the  center  gf,  the  required  equation 
(Art.  51,  I,  Cor.  1)  will  be 

(x-gy  +  (y-fy  =  r\ 
After  expansion  and  reduction,  this  assumes  the  form 

2fy  +  (g2  +f2  —  r2)  =  0: 


EQUATION  TO  CIRCLE.  139 

which  is  evidently  a  particular  case  of  the  general  equa- 
tion 

Ax2  +  ZHxy  +  By2  +  2Gx  +  ZFy  +  C=  0. 

Remark  —  The  annexed  diagram  will  ren- 
der clearer  the  geometric  meaning  of  the 
equation  just  obtained.  In  this,  we  have 
OM=x,  MP=y;  OM'=g,  M'C=f; 
and  CP  =  r.  Now,  drawing  PQ  parallel 
to  OX,  we  obtain  by  the  Pythagorean  theo- 
rem PQ2  +  QC*  =  CP\  That  is, 


13|>«    Equation   to   tbe   Circle,   axes   being   oblique.  _  To 

obtain  this,  we  have  simply  to  express,  in  terms  proper  for  oblique 
axes,  the  fact  that  the  distance  from  xy  to  gf  is  constant.  Hence, 
(Art.  51,  I,)  the  equation  is 


or,  in  the  expanded  form, 

*2+  2xy  cos  a)  +y2—  2  (g  +/cos  w)x—  2  (f+gcosw)y+(gi+2gfcos,<>)+f'2—  r2)=0. 


Equation  to  the  Circle,  referred  to  rec- 
tangular axes  witn  tne  Center  as  Origin.  —  Since 
the  equation  of  Art.  134  is  true  for  any  origin,  we  have 
only  to  suppose  in  it  g  =  0  and  /  =  0,  and  the  equation 
now  sought  is 


Remark  —  The  student  will  recognize  this  as  the  equation  of  Art. 
25,  II.  Its  great  simplicity  commends  it  to  constant  use.  It  may 
also  be  written 


a  form  analogous  to  that  of  the  equation  to  a  right  line, 


140  ANALYTIC   GEOMETRY. 

137.  Equation  to  the  Circle,  referred  to  any 
Diameter  and  the  Tangent  at  its  Vertex. — Since 
the  diameter  of  a  circle  is  perpendicular  to  its  vertical 
tangent,  in  order  to  obtain  the  present  equation  we  have 
merely  to  transform  the  last  one  to  parallel  axes  through 
( —  r,  0).     Writing,  then,  x  —  r  for  x  in  x2  +  2/2  =  r2,  and 
reducing,  we  find 

#2  +  y2  —  %rx  =  0  ; 
or, 

y2  =  2rx  —  x2. 

Kemark — In  the  equation  just  obtained,  we  suppose  the  origin 
to  be  at  the  left-hand  vertex  of  the  diameter.     It  is  customary  to 
adopt  this  convention.     If  the  origin  were  at  the  right-hand  vertex, 
we  should  have  to  replace  the  x  of  x2  +  y'2  =  r2  by 
x  -f  r,  and  the  equation  would  be 

z2  +  if  +  2 rx  =  0;  or.  y2  =  —  (2  rx  +  x*}. 

The  equation  of  this  article  is  verified  by  the  o 
diagram.     For    (Geom.,    325)    MF*  =  OM.MV; 
that  is, 

y1  s=  x  (2r  —  x)  =  2rx  —  x2. 

The  form  of  this  equation  (Art.  63)  shows  that  the  origin  is  on  the 
curve :  which  agrees  with  our  hypothesis. 

138.  Polar  Equation  to  the  Circle. — The  property 
that  the  distance  from  the  center  (d,  «)  to  the  variable 
point  of  the  curve  is  constant,  when  expressed  in  polars, 
(Art.  51,  II)  gives  us 

p2  +  d2  —  2pd  cos  (0  —  a)  ==  r2. 
Hence,  the  equation  now  sought  is 

p2  —  2tod  cos  (0  —  a)  =  r2  —  d2. 

Corollary  1. — Making  a  =  0  in  the  foregoing  expres- 
sion, we  obtain 

p2  —  2pd  cos  6  =  r2  —  d2 : 


POLAR  EQUATION  TO  CIRCLE.  141 

the  polar  equation  to  a  circle  whose  center  is  on  the  initial 
line. 

Corollary  2 — Making  d  =  0,  we  obtain  p2  =  r2',  or, 

p  =  constant : 
the  polar  equation  to  a  circle  whose  center  is  the  pole. 

Remark.— We  may  verify  these  equations 
geometrically  as  follows :  Let  OX  be  the 
initial  line,  and  C  the  center  of  the  circle. 


and  CP  =  r.     But  by  Trig.,  865,  OP2  +  OC2 
-  2  OP  .  0  C  cos  COP  =  CP\     Hence, 

p2  —  2pd  cos  (6  —  a)  ==  r2—  d*. 

If  the  point  C  fell  on  OX,  COP  would  become  XOP;  and  we 
should  have 


If  C  coincided  with  0,  OP  would  become  CP  ;  and  we  should  have 
p2_-?.2.  or^  p  =  constant 

These  equations  all  imply  that  for  every  value  of  6 
there  will  be  two  values  of  p  :  which  is  as  it  should  be, 
since  it  is  obvious  from  the  diagram  that  the  radius 
vector  OP,  corresponding  to  any  angle  XOP,  cuts  the 
curve  in  two  points,  P  and  P'. 

EXAMPLES. 

1.  Form  the  equation  to  the  circle  whose  center  is  (3,  4),  and 
whose  radius  =  2. 

2.  Lay  down  the  center  of  the  circle  (a;  —  2)2  -f  (y  —  6)2  =  25, 
and  determine  the  length  of  its  radius. 

3.  Do  the  same  in  the  case  of  the  circle  (*+  2)2+  (y  —  5)2  =  1; 
—  in  the  case  of  the  circle  x*  +  y2  =  3. 

4.  Form  the  equation  to  the  circle  whose  center  is  (5,  —  3)  and 
whose  radius  =1/7,  when  «=:600. 


142  ANALYTIC  GEOMETRY. 

5.  Form  the  equation  to  the  circle  whose  radius  =  3,  and  whose 
center  is  (3,  0).     Transform  the  equation  to  the  opposite  vertex  of 
the  diameter. 

6.  Form  the  equation  to  the  circle  whose  radius  =  6,  and  whose 
center  lies,  at  a  distance  from  the  pole  =  5,  on  a  line  which  makes 
with  the  initial  line  an  angle  =  60°. 

7.  Form  the  equation  to  the  circle  whose  center  is  on  the  initial 
line  at  a  distance  of  16  inches  from  the  pole,  and  whose  radius  = 
1  foot. 

8.  Transform  the  equation  of  Ex.  6  to  rectangular  axes,  the  origin 
being  coincident  with  the  pole. 

9.  Form  the  equation  to  the  circle  whose  center  is  at  the  pole, 
and  whose  radius  =  3.     Transform  it  to  rectangular  axes,  origin 
same  as  pole. 

10.  Form  the  equation  to  an  infinitely  small  circle  whose  center 
is  (a,  0).  [Compare  the  result  with  the  Remark,  Art.  61,  and  the 
Note,  Art.  127.] 

II.    ANALYTIC    POINT    OF   VIEW:  —  THE    EQUATION    OF    THE 

SECOND    DEGREE    ON   A    DETERMINATE    CONDITION 

REPRESENTS   A   CIRCLE. 

189.  The  theorem  is  implied  in  the  fact  established 
in  Arts.  134,  135,  that  the  equation  to  the  Circle  is  a 
particular  case  of  the  general  equation  of  the  second 
degree.  To  determine  the  condition  on  which  the  gen- 
eral equation  will  represent  a  circle,  we  have  therefore 
merely  to  compare  its  co-efficients  with  those  of  the 
equation  to  the  Circle  written  in  its  most  general  form. 

I4O.  We  saw  (Art.  134)  that  the  rectangular  equation 
to  the  Circle  is 


Since  g  and  f  may  be  either  positive  or  negative,  and  r 
is  not  a  function  of  either  /  or  g,  this  may  be  written  in 
the  still  more  general  form 

A  (x2  +  if)  +  2fo 


CIRCLE  AND  GENERAL  EQUATION.  143 

If,  then,  the  equation 

Ax1  +  ZHxy  +  By*  +  2Gx  +  2Fy  +  C=  0 

represent  a  circle,  it  must  assume  the  form  just  obtained. 
But  in  order  to  this,  we  must  have 

H'=  0  with  A  =  B  : 

which  therefore  constitutes  the  condition  that  the  general 
equation  of  the  second  degree  shall  represent  a  circle. 

Corollary.  —  It  is  obvious  from  the  condition  just  deter- 
mined, that  when  the  equation  of  the  second  degree 
represents  a  circle,  it  also  fulfills  the  condition 

ff2  —  AB<0. 

141.   From  the  result  of  Art,  135,  it  follows  that 
H  =  A  cos  a  with  A  =  B 

is  the  condition  in  oblique  axes  that  the  equation  of  the  second  degree 
shall  represent  a  circle.  And  it  is  evident  that  in  this  case,  too,  we 
have  the  condition 


142.  If  we   are  given   an   equation   in   the    general 
form 


we  can  at  once  determine  the  position  and  magnitude  of 
the  corresponding  circle. 

For,  by  transposing  (7,  adding -7 to  both  mem- 
bers, and  dividing  through  by  A,  the  given  equation  may 
be  thrown  into  the  form 

£2  +FZ  —  AC 


A2 
Comparing  this  with  the  equation  of  Art.  134,  namely, 


144  ANALYTIC  GEOMETRY. 

we  obtain,  for  determining  the  co-ordinates  of  the  center, 

G  F 

9-    -3'    /=    -3' 

and  for  determining  the  radius, 


r  = 


A 


Corollary  1.  —  From  the  expressions  for  g  and  /,  which 
are  independent  of  (7,  we  learn  the  important  principle, 
that  the  equations  to  concentric  circles  differ  only  in  their 
constant  terms. 

Corollary  2.  —  From  the  expression  for  r,  we  derive  the 
followin  conclusions  : 


I.   G2  +  F2  —  AC>0  .'.  the  circle  is  real 
II.   G2  -f-  F2  —  AC=  0  /.  the  circle  is  infinitely  small. 
III.  G2-\-F2  —  AC<Q  .-.  the  circle  is  imaginary. 

143.  We  may  also  determine  the  position  and  magni- 
tude of  the  circle  by  finding  its  intercepts  on  the  two 
axes. 

Thus,  if  we  make  the  y  and  x  of  the  given  equation 
successively  =  0,  we  obtain  the  two  equations 


Ax1  +  2Gx  +  C=^  0,      Ay2  +  2Fy  +  0=  0. 


Since  each  of  these  is  a  quadratic,  the  circle  cuts  each 
axis  in  two  points;  and  as  a  circle  is  completely  deter- 
mined by  three  points,  its  center  and  radius  are  a  fortiori 
fixed  by  the  four  points  thus  found.  We  have  therefore 
only  to  find  the  co-ordinates  of  the  point  equidistant  from 
either  three  of  the  four,  and  we  obtain  the  center  :  the 
distance  between  this  and  any  one  of  the  four,  is  the  ra- 
dius. Or  we  may  proceed  with  greater  brevity  as  follows  : 


CIRCLE  TOUCHING  AXES.  145 

Let  the  intercepts  on  the  axis  of  x  be  x1 ',  x" ;  and  those 
on  the  axis  of  y  be  ?/',  y" :  then  (Alg.,  234,  Props.  3d 
and  4th)  2G:  A=  ->'  -f  a/'),  2jF  :  ^L  =  -  (yf  +  y"), 
and  (7  :  J.  =  a/o"  =  #y .  Hence,  (Art.  142,)  for  deter- 
mining the  center, 

_x>_  +  x"  y'  +  y»m 

CJ~         2  I"          2 

and  for  determining  the  radius, 


r  =  \  V  (x'*^  x"2}  -{-  (y12  +  y"2). 

Corollary.  —  Hence,  the  equation  to  any  circle  whose 
intercepts  on  the  axis  of  x  are  given,  is 

x'2  +  y2—  (X  +  *")  *  —  2/y  +  v'x"  =  o  ; 

the  equation  to  any  circle  whose  intercepts  on  the  axis 
of  y  are  given,  is 

x-'  +f-  2gx  -  (y'  +  */")  y  +  y'y"  =  0  ; 

and  the  equation  to  the  circle  whose  intercepts  on  both 
axes  are  given,  is 

2  (x^f]  -  2  (x'+x")  x-2  (y'+y")  y+(x'x"+y'y"}  =  0. 
144.  Of  the  two  equations 


the   first   (Alg.,   237,   1)    will   have    equal    roots    when 
G2-=AC;  and  the  second,  when  JF2  =  AC.     Hence, 

G2  —  AC=Q 
is  the  condition  that  a  circle  shall  touch  the  axis  of  x  ; 

F2—  AC=  0 
is  the  condition  that  it  shall  touch  the  axis  of  j  ;  and 


is  the  condition  that  it  shall  touch  both  axes. 


146  ANALYTIC   GEOMETRY. 

EXAMPLES. 

I.    NOTATION    AND    CONDITIONS. 

1.  Decide  whether  the  following  equations  represent  circles : 

3.?2  +  5xy  —  ly1  +  2x  —  4y  +  8  =  0 ; 

5,-r2  —  Sy8  +  3x  —  2y  +  7  =  0 ; 
5z2  -1-  5y2  —  IQx  —  30y  +  15  =  0. 

2.  Determine  the  center  and  radius  of  each  of  the  circles 

z2+7/2  +  42/-4a;-l=0, 
z2  -f  7/2  +  6x  —  3y  —  1  =  0. 

3.  Form  the  equation  to  the  circle  which  passes  through  the 
origin,  and  makes  on  the  two  axes  respectively  the  intercepts  -f-  h 
and  -f-  k- 

4.  Find  the  points  in  which  the  circle  x'2^y'2  =  3  intersects 
the  lines  x+y  +  1=  0,  x+y  —  1=0,  and  2x  +  y  1/5=9. 

5.  What  must  be  the  inclination  of  the  axes  in  order  that  each 
of  the  equations 

xz  —  xy  -f  «/'2  —  Aa;  —  hy  =  0, 
x2  -f-  xy  -{-  y2  • —  A  a;  —  Ay  =  0, 

may  represent  a  circle  ?     Determine  the  magnitude  and  position 
of  each  circle. 

6.  Write  the  equations  to  any  three  circles  concentric  with 

2(s2  +  y2)  +  6x  —  4y—  12  =  0. 

7.  Form  the  equation  to  the  circle  which  makes  on  the  axis 
of  x  the  intercepts  (5,  —  12),  and  on  the  axis  of  y  the  intercepts 
(4,  — 15).     Determine  the  center  and  radius  of  the  same. 

8.  What  is  the  equation  to  the  circle  which  touches  the  axes 
at  distances  from  the  origin,  each  =  a  ?  —  at  distances  =  5  and  6 
respectively  ? 

9.  ABC  is  an  equilateral  triangle:  taking  A  as  pole,  and  AB 
as  initial  line,  form  the  polar  equation  to  the  circumscribed  circle. 
Transform  it  to  rectangular  axes,  origin  same  as  pole,  and  axis  of  x 
as  initial  line. 

10.  If  S  —  0  and  S/  =  0  are  the  equations  to  any  two  circles, 
what  does  the  equation  S —  P/S"  =  0  represent,  k  being  arbitrary  ? 


EXAMPLES  ON  THE  CIRCLE. 


147 


II.    CIRCULAR   LOCI. 

1.  Given  the  base  of  a  triangle  and  the  sum  of  the  squares  on 
its  sides  :  to  find  the  locus  of  its  vertex. 

Taking  the  base  and  a  perpendicular  through 
its  middle  point  for  axes,  putting  2s2  =  the  given 
sum  of  squares,  and  in  other  respects  using  the 
notation  of  Ex.  1,  p.  126,  we  have 

PR2  =  3/2  +  (m  +  a-)2,      PQ*  =  y2  +  (m  -  xp. 
Hence,  the  equation  to  the  required  locus  is 


M      Q 


and  the  vertex  moves  upon  a  circle  whose  center  is  the  middle  point  of  the 
base,  and  whose  radius  =  V«2  —  m'z. 

2.  Given  the  base  and  the  vertical  angle  of  a  triangle  :  to  find 
the  locus  of  the  vertex. 

3.  Given  the  base  and  the  vertical  angle  of  a  triangle  :  to  find 
the  locus  of  the  center  of  the  inscribed  circle. 

4.  Find  the  locus  of  the  middle  points  of  chords  drawn  from 
the  vertex  of  any  diameter  in  a  circle. 

5.  Given  the  base  and  the  ratio  of  the  sides  of  a  triangle:  to 
find  the  locus  of  the  vertex. 

6.  Given  the  base  and  vertical  angle:  to  find  the  locus  of  the 
intersection  of  the  perpendiculars  from  the  extremities  of  the  base 
to  the  opposite  sides. 

7.  When  will  the  locus  of  a  point  be  a  circle,  if  the  square 
of  its  distance  from  the  base  of  a  triangle  bears  a  constant  ratio 
to  the  product  of  its  distances  from  the  sides  ? 

8.  When  will  the  locus  of  a  point  be  a  circle,  if  the  sum  of  the 
squares  of  its  distances  from  the  three  sides  of  a  triangle  is  con- 
stant ? 

9.  ABC  is  an  equilateral  triangle,  and  P  is  a  point  such  that 

PA  =  PB  +  PC: 
find  the  locus  of  P. 

10.  A  C  B  is  the  segment  of  a  circle,  and  any  chord  A  C  is  pro- 
duced to  a  point  P  such  that  AC—  nCP  :  to  find  the  locus  of  P. 


148  ANALYTIC  GEOMETRY. 

11.  To  find  the  locus  of  the  middle  point  of  any  chord  of  a  circle, 
when  the  chord  passes  through  any  fixed  point. 

12.  On  any  circular  radius  vector  OQ,  OP  is  taken  in  a  constant 
ratio  to  OQ :  find  the  locus  of  P. 

13.  Find  the  locus  of  P,  the  square  of  whose  distance  from  a 
fixed  point  O  is  proportional  to  its  distance  from  a  given  right  line. 

14.  O  is  a  fixed  point,  and  AS  a  fixed  right  line;  a  line  is  drawn 
from  O  to  meet  AB  in  Q,  and  on  OQ  a  point  P  is  taken  so  that 
OQ.OP=  k* :  to  find  the  locus  of  P. 

15.  A  right  line  is  drawn  from  a  fixed  point  O  to  meet  a  fixed 
circle  in  Q,  and  on  OQ  the  point  P  is  so  taken  that  OQ.OP=  k2: 
to  find  the  locus  of  P. 


SECTION  V.  —  THE  ELLIPSE. 

I.    GEOMETRIC    POINT    OF   VIEW  I — THE    EQUATION    TO    THE 
ELLIPSE    IS    OF    THE    SECOND    DEGREE. 

145.  The  Ellipse  may  be  defined  by  the  following 
property :  The  sum  of  the  distances  from  the  variable 
point  of  the  curve  to  two  fixed  points  is  constant. 

We  may  therefore   trace  the  curve  and  discover  its 
figure  by  the  following  process  :  —  Take  any  two  points 
F'  and  F,  and  fasten  in  them  the 
extremities  of  a  thread  whose  length 
is    greater    than    F'F.      Place    the 
point    of    a    pencil    P   against    the 
thread,   and   slide  it  so   as  to  keep 
the  thread  constantly  stretched:   P 
in  its  motion  will   describe   an   ellipse.     For,  in   every 
position  of  P,  we  shall  have 

F'P+FP=  constant, 

as  the  sum  of  these  distances  will  always  be  equal  to  the 
fixed  length  of  the  thread. 


EQUATION  TO  ELLIPSE. 


149 


146.  The  two  fixed  points,  F'  and  F,  are  called  the 
foci;  and  the  distances  with  a  constant  sum,  F'P  and 
FP,  the  focal  radii  of  the  curve. 

The  right  line  drawn  through  the 
foci  to  meet  the  curve  in  A!  and  A, 
is  called  the  transverse  axis.  The  A, 
point  0,  taken  midway  between  F' 
and  F,  we  may  for  the  present  call 
the  focal  center. 

The  line  B'B,  drawn  through  0  at  right  angles  to 
A' A,  and  terminated  by  the  curve,  is  called  the  conjugate 
axis. 

147.  Equation  to  the  Ellipse,  referred  to  its 
Axes. — Let  %c  =  the  constant  distance  between  the  foci, 
and  2a  =  the  constant  sum  of  the  focal  radii. 

Then,  from  the  diagram  above,  F'P2  =  (x-\-  c)2  -f-  ?/2, 
and  FP2  =  (x  —  c)2  -f-  yz.  Hence,  the  fundamental  prop- 
erty of  the  Ellipse,  expressed  in  algebraic  symbols, 
will  be 


Freeing   this   expression   from   radicals,   we   obtain   the 
required  equation, 

(a2  —  c2)  x2  +  a2if  =  a2  (a2  -  c2)  (1) . 
To  abbreviate,  put  a2  —  c2  =  b2,  and  this  becomes 

b2x2  +  ay  =  a2b2  (2): 
which  may  be  more  symmetrically  written 

^    .   2/!  —  -, 
a2~r  52  —  1. 


Eemark — The  student  will  observe  the  analogy  between  the  last 
form  and  the  equation  to  the  Right  Line  in  terms  of  its  intercepts. 


150  ANALYTIC  GEOMETRY. 

148.  It  is  important  that  we  should  get  a  clear  con- 
ception of  the  general  form  of  the  equation  just  obtained. 
Let  us  for  a  moment  return  to  the  form  (1), 

(a2  —  c2)  x2  +  ay  =  a2  (a2  —  c2). 

In  this,  the  definition  of  the  Ellipse  requires  that  a2  —  c2 
shall  have  the  same  sign  as  a2 ;  for  the  sura  of  the  dis- 
tances of  a  point  from  two  fixed  points  can  not  be  less 
than  the  distance  between  them :  that  is,  a  can  not  be 
numerically  less  than  c,  and  consequently  a2  not  less  than 
c2.  Therefore,  in  the  equation  to  the  Ellipse,  the  co-effi- 
cients of  XL  and  y1  must  have  like  signs. 
Advancing  now  to  the  form  (2), 

b2x2  -f-  a2y2  —  a2b2, 

it  is  obvious  that  the  constant  term  a2b2  will  have  but  one 
sign,  whether  the  co-efficients  of  x2  and  y2  be  both  positive 
or  both  negative.  Supposing,  then,  that  a2  and  b2  are  both 
positive,  the  equation  after  transposition  would  be 

b2x*-\-  a2f  —  a*b2  =  Q  (3). 

Supposing  them  both  negative,  it  would  become,  after 
transposition  and  the  changing  of  its  signs, 

b2x2  +  a2y2  +  a2b2  =  0  (4). 

Now,  what  is  the  meaning  of  the  supposition  that  a2 
(and  thence  b2)  is  negative  ?  Plainly  (since  in  that  case 
we  shall  have,  instead  of  a,  ai/--l)  it  signifies  that  the 
corresponding  ellipse  is  imaginary*  Hence,  admitting 
into  our  conception  of  this  curve  the  imaginary  locus 
of  (4),  we  learn  that  the  equation  to  the  Ellipse  is  of  the 
general  form 

A'x2  +  B'y2  +  C1  =  0  : 


*  This  interpretation  is  put  beyond  question  by  the  form  of  equation 
(4)  itself,  which  denotes  an  impossible  relation. 


POLAR  EQUATION  TO  ELLIPSE.  151 

in  which  A!  and  Bf  are  positive,  and  Cr  is  either  positive 
or  negative  according  as  the  curve  is  imaginary  or  real. 

149.  We  may  at  this  point  derive  from  the  equation 
to   the  Ellipse   a   single   property   of  the   curve,  as  we 
shall  need  it  in  discussing  the  general  equation  of  the 
second  degree. 

Definition. — A  Center  of  a  Curve  is  a  point  which 
bisects  every  right  line  drawn  through  it  to  meet  the 
curve. 

Theorem, — In  any  ellipse,  the  focal  center  is  the  center 
of  the  curve.  The  equation  to  any  right  line  drawn 
through  the  focal  center  (Art.  78,  Cor.  5)  is 

y  =  mx. 
Comparing  this  with  the  equation  to  any  ellipse,  namely, 

x2       y2 

^  +  ^  =  :1' 

we  see  that  if  xf,  y'  satisfy  both  equations,  —  #',  —  yr 
will  also  satisfy  both.  In  other  words,  the  points  in 
which  the  ellipse  is  cut  by  any  right  line  drawn  through 
the  focal  center  may  be  represented  by  xr,  y'  and  —  xf, 
—  yr.  But  these  symbols  (Art.  51,  I,  Cor.  2)  necessarily 
denote  two  points  equidistant  from  the  focal  center : 
which  proves  the  proposition. 

150.  Polar  Equation  to  the  Ellipse,  the  Center 
being  the  Pole. — Replacing  (Art.  57,  Cor.)  the  x  and 
y  of  a2y2  +  b2x*  =  a2b2  by  p  cos  6  and  p  sin  6,  we  find 

97  ? 

2 a~fr 

"       az  sin2#  -j-  b2  co&26  ' 
that  is,  (Trig.,  838,) 

9_  a2b2 

p"       a2—  (a2—  b2)  cos20  ' 
An.  Ge.  16. 


152  ANALYTIC  GEOMETRY. 

Divide  both  terms  of  the  second  member  of  this  expres- 
sion by  a2,  and,  to  abbreviate,  put 


the  result  will  be  the  form  in  which  the  required  equation 
is  usually  written,  namely, 

v 

P    —  i  -  9  -  Ta  ' 

1  —  e2  cos2d 

Remark.  —  The  equation  indicates  that  for  any  value  of  0  there 
will  be  two  radii  vectores,  numerically 
equal  with  contrary  signs.  The  accom- 
panying diagram  verifies  this  result;  for 
the  two  points  of  the  ellipse,  P  and  P1  ', 
evidently  correspond  to  the  same  angle  0, 
and  the  point  P'  has  the  radius  vector 
OP'  =  —  OP. 

151.  Special   attention   should  be  given  to  the  two 
abbreviations  used  above, 

2          12  ,1       ^—V 

a?  —  c2  =  b2    and    -  —  =  e- 

We  shall  find  hereafter  that  they  represent  elements  of 
great  significance  in  the  Conies.  For  the  present,  how- 
ever, they  are  to  be  regarded  as  abbreviations  merely. 
It  is  evident  that  by  combining  them  we  obtain  the 
relation 

c  =  ae. 

Corollary,  —  Hence,  the  central  polar  equation  to  the 
Ellipse  may  be  written 


n2  —  _ 

~    ~    — 


a2  (1  —  e2) 


a  form  which  will  frequently  prove  more  convenient  than 
that  of  Art.  150. 


POLE  AT  FOCUS.  153 

152.  Polar  Equation  to  the  Ellipse,  the  Focus 
being  the  Pole.  —  From  the  annexed  diagram,  we  have 
F'P  =  p,  and  FP  =  y(p2  +  4c2  —  4/w  cos  6).  Hence, 
expressing  the  fundamental  property 
of  the  Ellipse, 


a2  —  c2 
p  =  - 
"          — 


c  cos 


Replacing  c  by  its  equal  ae,  we  obtain  the  usual  form 
of  the  equation, 


"    1  —  g  cos  # 

Remark.  —  The  student  should  carefully  discriminate  between 
this  equation  and  that  of  the  corollary  to  Art.  151.  From  their 
striking  similarity,  the  two  are  liable  to  be  confounded. 

In  using  this  equation,  it  is  to  be  remembered  that  in  it  the  left- 
hand  focus  is  taken  for  the  pole.  In  practice,  this  assumption  is 
generally  found  to  be  the  more  convenient.  The  student  may  show, 
however,  that  when  the  right-hand  focus  is  the  pole  the  equation  to 
the  Ellipse  is 


p  = 


1  -f  e  cos  0 


EXAMPLES. 

1.  Given  the  two  points  ( — 3,  0)  and  (3,  0);   the  extremities 
of  a  thread  whose  length  =10  are  fastened  in  them:  form  the 
equation  to  the  ellipse  generated  by  pushing  a  pencil  along  the 
thread  so  as  to  keep  it  stretched. 

2.  In  a  given  ellipse,  half  the  sum  of  the  focal  radii  =  3,  and 
half  the  distance  between  the  foci  =  2 :  write  its  equation. 

3.  Form  the  equation  to  the  ellipse  whose  focus  is  3  inches 
from  its  center,  and  whose  focal  radii  have  lengths  whose  constant 
sum  =  1  foot. 

4.  In  a  given  ellipse,  the  sum  of  the  focal  radii  =  8,  and  the 
difference  between  the  squares  of  half  that  sum  and  half  the  dis- 
tance between  the  foci  =  9 :  write  its  equation. 


154  ANALYTIC  GEOMETRY. 

5.  Show  that  in  each  of  the  ellipses  hitherto  given,  the  focal 
center  bisects  the   lines  y  =  2#,  y  —  3#,  y  =  5#,  and  any  others 
whose  equations  are  in  the  form  y  =  mx. 

6.  Write  the  central  polar  equation  to  the  ellipse  in  which  the 
difference  between  the  squares  of  half  the  sum  of  the  focal  radii 
and  half  the  distance  between  the  foci  —  9,  and  the  ratio  of  the 
distance  between  the  foci  to  the  sum  of  the  focal  radii  =  1:2. 

7.  "Write  the  central  polar  equation  to  the  ellipse  of  Ex.  2. 

8.  Find,  in  the  same  ellipse,  the  ratio  of  the  sum  of  the  focal 
radii  to  the  distance  between  the  foci,  and  write  the  equation  in 
the  form  corresponding  to  that  in  the  corollary  to  Art.  151. 

9.  In  a  given  ellipse,  the  sum  of  the  focal  radii  =  12,  and  the 
ratio  between  that  sum  and  the  distance  from  the  left-hand  focus 
to  the  right-hand  one  =  3 :  write  its  polar  equation,  the  focus  being 
the  pole. 

10.  The  focus  being  the  pole,  form  the  polar  equation  to  the 
ellipse  of  Ex.  3.  What  would  the  equation  be,  if  the  right-hand 
focus  were  the  pole  ? 


II.    ANALYTIC    POINT   OF   VIEW:  —  THE    EQUATION    OF    THE 

SECOND   DEGREE    ON   A   DETERMINATE    CONDITION 

REPRESENTS   AN   ELLIPSE. 

153.  We  have  seen  (Art.  148)  that  the  equation  to 
the  Ellipse  may  always  be  written  in  the  form 

A'x2  +  B'y2  +  Cr  =  0, 

in  which  A'  and  B'  are  positive,  and  Cr  is  indeterminate 
in  sign.  If,  then,  we  can  show  that  the  general  equation 
of  the  second  degree  is  reducible  to  this  form,  and  can 
find  real  conditions  upon  which  the  reduction  may  always 
be  effected,  we  shall  have  established  the  theorem  at  the 
head  of  this  article. 


ELLIPSE  AND  GENERAL  EQUATION.  155 

154.  In    order    that    the    general    equation    of    the 
second  degree, 


Ax*  +  2Hxy  +  By2  +  2Gx  +  ZFy  +  C=  0, 
may  assume  the  form 

A''*2  +  Bfy2  +  C'  =  0, 

the  co-efficients  of  its  x,  y,  and  xy  must  all  vanish. 
The  question  therefore  is  :  Can  we  transform  the  equation 
to  axes  such  as  will  cause  these  co-efficients  to  disappear  ? 

155.  As  we  have  seen,  the  equation  Arx2jrB'y2-\-  Cf=0 
is  referred  to  the  center  of  the  Ellipse,  and  to  its  axes, 
which  by  definition  cut  each  other  at  right  angles.  As- 
suming, then,  as  we  may,  that  the  general  equation  of 
the  second  degree  as  above  written  is  referred  to  rectan- 
gular axes,  our  first  step  will  naturally  be  to  determine, 
if  possible,  the  center  of  the  locus  which  it  represents,  and 
to  reduce  it  to  that  center  as  origin. 

Let  xr,  y'  be  the  co-ordinates  of  the  center  sought, 
and  let  us  transform 


Ax2  +  2ffxy  -f  By2  +  2Gx  -j-  2Fy  +  tf=  0          (1) 

to  parallel  axes  passing  through  x'y'  .  Replacing  (Art.  55) 
the  x  and  y  of  (1)  by  (xr  -j-  x)  and  (yf  -f  y),  we  obtain, 
after  reductions, 

Ax2  -f  2Hxy  +  By2  \ 


-f  Axf2  -f  ZHx'y'  +  By'2  +  2Gxf  +  2Fyr  +  C  ) 

Now,  since  this  equation  is  referred  to  the  center  as 
origin,  it  must  (Art.  149)  be  satisfied  equally  by  x,  y 
and  —  x,—y.  But  in  order  to  this,  we  must  have 


156  ANALYTIC  GEOMETRY. 

Eliminating  between  these  simultaneous  equations,  we 
find  the  co-ordinaies  of  the  center, 

BG—HF       f      AF—ffG 
~  E*-AB  '  y  ~~-  R2  —  AB  ' 

It  is  obvious  that  these  values  of  xf  and  y'  will  be  finite  so 
long,  and  only  so  long,  as  H2  —  AB  is  not  equal  to  zero. 
Hence  we  conclude  that  the  locus  of  (1)  has  a  center, 
which  is  situated  at  a  finite  distance  from  the  origin  or 
at  infinity,  according  as  (1)  does  not  or  does  fulfill  the 
condition  H2  —  AB  =  Q. 

If,  then,  in  the  result  of  our  first  transformation  above, 
we  substitute  these  values  of  x'  and  y1  ',  we  shall  obtain 
an  equation  to  the  locus  of  (1),  referred  to  its  center. 
Now  the  only  elements  of  that  result  which  depend  on 
xf  and  y',  are  the  co-efficients  of  x  and  y,  and  the  abso- 
lute term.  Of  these,  the  first  two  vanish,  when  the  finite 
co-ordinates  of  the  center  are  substituted  in  them  ;  the 
third  may  be  thrown  into  the  form 

{Axf+Htf+G)  *'+  (By'  +  Hx'+F]  y>+  (Gx'+Fy'+C}  : 

and  if  in  this  we  substitute  the  finite  co-ordinates  of  the 
center,  it  becomes 


or, 

ABC+2FGH—AF2-BG2-CH* 
H2  —  AB 

Hence,  putting  Cf  to  represent  either  (a)  or  (5),  the 
equation  of  the  second  degree,  reduced  to  the  geometric 
center  of  its  locus,  is 

'=Q  (2). 


ELLIPSE  AND  GENERAL  EQUATION.  157 

156.  The  transformation  from  (1)  to  (2)  has  destroyed 
the  co-efficients  of  x  and  y,  but  the  co-efficient  of  xy  still 
remains.  Reduction  to  the  center  as  origin  is  therefore 
not  sufficient  to  bring  (1)  into  the  form 


And,  in  fact,  we  might  have  anticipated  as  much;  for 
the  equation  to  the  Ellipse,  of  which  the  required  form 
is  the  type,  is  referred  not  to  the  center  merely,  but  to 
the  axes  of  the  curve.  To  destroy  the  co-efficient  of  xy, 
then,  we  must  resort  to  additional  transformation  ;  and 
our  next  step  will  naturally  be  to  determine,  if  possible, 
the  axes  of  the  locus  represented  by  (2),  and  to  revolve  the 
reference-axes  until  they  coincide  with  them. 

Let  6  =  the  angle  made  with  the  reference-axes  of 
(2)  by  the  possible  axes  of  the  locus.  Replacing  (Art. 
56,  Cor.  3)  the  x  and  y  of  (2)  by  x  cos  6  —  y  sin  0  and 
x  sin  6  -f  y  cos  0,  we  obtain,  after  reductions, 

(A  cos2  d-\-2H  sin  dcosd  +  B  sin2  0)  x2     ^ 
-  2  {  (A—B)  sin  6  cos  6—H(cos2  6—  sin2  d)}xy  I  +  C'=Q  ; 
-|-  (A  sin2  6  —  2#sin  dcosd  +  B  cos2  6)  y2     J 

that  is,  (Trig.,  847  :  I,  n,  IV,) 

-\ 

L+C"=0. 
+  J  {(A+B)—  (A—£)cos20—2ffsm2d}y2  ) 

If,  in  this  expression,  we  equate  to  zero  the  co-efficient 
of  xy,  we  shall  have 

(A  —  B]  sin  20  —  2#cos  20  =  0  : 
.-.  tan  20  = 


J?) 


158  ANALYTIC  GEOMETRY. 

That  is,  since  a  tangent  may  have  any  value  positive  or 
negative  from  0  to  GO,  20  (and  therefore  6)  is  a  real  angle  ; 
in  other  words,  there  do  exist  two  real  lines,  at  right  angles 
to  each  other,  which  in  virtue  of  their  destroying  the  co- 
efficient of  xy  we  may  call  axes  of  the  curve  to  the  locus 
of  (2).  Accordingly,  if  in  the  equation  last  obtained  we 
substitute  for  the  functions  of  2#  their  values  as  implied 
in  (c),*  the  resulting  equation  will  represent  the  locus, 
referred  to  these  so-called  axes. 


cos  9#  —  " 


Substituting  these  values  in  our  last  equation,  we  find 

„, 

~ 


+  J  {  (A  +  B)  -  i/(A  — 
whence,  by  writing 


=  J  {(A  +  B)  +  l/p-J?)2  +  (2#)2}        (d), 

(e), 


the  equation  of  the  second  degree,  reduced  to  the  axes  of 
its  locus,  is 

4V  +  £y  +  C"  =  o  (3). 


*By  Trig.,  836,  sin  A  =  -;    cos  A  =  -  .      But  since  c  represents   the 

hypotenuse,  and  a  and   b   the   sides,  of  a   right  triangle:   c  =  Va*  +  b*~. 

a 
~b 


Hence,  when  the  tangent  is  given,  e.  g.  tan  A  =  j- ,  we  at  once  derive  the 


sine  and  cosine  by  writing 


CRITERION  OF  THE  ELLIPSE.  159 

From  (3)  it  appears,  that,  setting  aside  the 
question  of  signs,  the  general  equation  of  the  second 
degree  can  be  reduced  to  the  required  form ;  provided  it 
is  not  subject  to  the  condition  H2-—AB  =  0. 

158.  It  remains,  then,  only  to  inquire  what  condition 
the  general  equation  must  fulfill  in  order  that  its  reduced 
form  (3)  may  have  that  combination  of  signs  which  (Art. 
148)  is  characteristic  of  the  Ellipse. 

If  A!  and  B1  are  both  positive,  we  shall  have 

A'B'  =  positive ; 

or,  by  substituting  for  A'  and  B'  from  (d)  and  (e)  above, 
and  reducing, 

AB  —  H2  =  positive  ; 

that  is,  changing  the  signs  of  both  sides  of  the  expression, 
If2  —  AB  =  negative. 

Hence,  The  equation  of  the  second  degree  represents  an 
ellipse  whenever  its  co-efficients  fulfill  the  condition 

H2—AB<0. 

1*>O.  At  the  close  of  Art.  148  we  saw  that  the  sign 
of  Cr  is  plus  or  minus  according  as  the  ellipse  is  imag- 
inary or  real.  Let  us  then  seek  the  conditions  which  the 
general  equation  must  fulfill  in  order  to  distinguish  between 
these  two  states  of  the  curve. 

Applying  the  condition  H2  —  AB  <  0  to  the  value 
of  G1  as  given  in  (b)  of  Art.  155,  we  see  that,  for  C'  to 
be  negative,  we  must  have 

ABC  +  2FGH—  AF2  —  BG2  —  CH2  <  0 ; 

and,  for  Cf  to  be  positive, 

ABC  +  2FGH—AF2  —  BO2  —  OH2  >  0. 

An.  Ge.  17. 


160  ANALYTIC  GEOMETRY. 

In  other  words,  we  find  that  the  same  quantity  which 
(Art.  131)  by  vanishing  indicates  a  pair  of  right  lines  as 
the  locus  of  the  general  equation,  by  changing  sign  indi- 
cates the  transition  from  the  real  to  the  imaginary  ellipse. 
This  quantity  is  called,  in  modern  algebra,  the  Discrimi- 
nant of  the  general  equation ;  and  we  may  appropriately 
represent  it  by  the  Greek  letter  J.  Adopting  this  nota- 
tion, we  have 

H*  —  AB<0  with  J<0 

as  the  condition  that  the  equation  of  the  second  degree  shall 
represent  a  real  ellipse;  and 

H2  —  AB<Q  with  J>0 
as  the  condition  that  it  shall  represent  an  imaginary  one.* 

16O.  We  can  now  see,  at  least  in  part,  the  real  bearing 
of  the  conditions  in  terms  of  H2  —  AB  which  we  some 
time  ago  developed  respecting  Pairs  of  Right  Lines. 

Comparing  the  results  of  Arts.  131,  132,  we  infer  that 

H2  —  AB<0  with  J  =  0  (7) 

is  the  condition  that  the  equation  of  the  second  degree 
shall  represent  two  imaginary  intersecting  lines.  But  this 
condition  evidently  lies  between  the  two  criteria 

ff2  —  AB  <  0  with  J '  <  0  (&), 

H2  —  AB<0  with  J>0  (m); 

so  that  we  can  not  pass  from  (k)  to  (m)  without  passing 
through  (7).  We  thus  learn  that  two  imaginary  right 
lines  intersecting  each  other,  form  the  limit  between  the 
real  and  the  imaginary  ellipse. 

If  we  now  revert  to  the  equation  (Art.  132)  denoting 


*  In  testing  any  given  equation  by  these  criteria,  we  must  see  that  its 
signs  are  so  arranged  that  A  (the  co-efficient  of  x2)  may  be  positive.  The 
conditions  with  respect  to  A,  are  derived  on  this  assumption. 


POINT  AND  CIRCLE  AS  ELLIPSE. 


161 


two  right  lines,  and  take  its  two  roots  separately,  we  see 
that  the  two  lines  are 


/  TT          _,  /  JJT'Z  /\    D  \  I       7?          I      /  Tjt          i  /  T/i'> £>  /Y   \   A 

(  XZ  K    ±1 -/l-O    )  iC  ~+~  .Dl/  —]~   \Ju |/  M  " />  L/     )   :::=:   U. 

Eliminating  between   these   equations,  and  recollecting 
[Art.  131,  (1)1  that 

(HF-BGY 


we  find,  as  the  co-ordinates  of  intersection  for  the  two  lines, 
BG  —  HF  AF—HG 


~ 


~  H2—AB' 

That  is,  (Art.  155,)  the  lines  intersect  in  the  center  of  the 
locus  of  the  general  equation.  But  we  have  seen  that  this 
center  is  real,  irrespective  of  the  state  of  H2  —  AB\  and 
is  finite,  so  long  as  H2  —  AB  is  not  zero.  Hence,  when- 
ever the  equation  of  the  second  degree  represents  two 
intersecting  lines,  their  intersection  is  a  finite  real  point, 
whether  they  be  real  or  imaginary. 

Uniting  the  two  conclusions  thus  reached,  we  obtain 
the  following  important  theorem  :  The  Point,  as  the  inter- 
section of  two  imaginary  right  lines,  is  the  limiting  case 
of  the  Ellipse. 

Remark,  —  This  result  is  corroborated  by  the  equation  (Art.  148) 
to  the  Ellipse  itself.     For  if,  in  the  expression 
'  A'x2  +  BY  +  C"  =  0, 
we  suppose  A  =  0,  then  C"  =  0  ,  and  the  equation  becomes 


which  (Art.  126,  Cor.  2  cf.  Art.  127)  denotes  two  imaginary  lines 
passing  through  the  origin  ;  that  is,  in  this  case,  through  the  center. 

161.  The  Point  and  the  Pair  of  Imaginary  Intersecting 
Lines  have  thus  been  brought  within  the  order  of  Conies. 
We  shall  now  show  that  the  Circle  likewise  belongs  there. 


162  ANALYTIC  GEOMETRY. 

The  condition  that  the  equation  of  the  second  degree 
shall  represent  a  circle  (Art.  140)  is 


But,  as  we  noticed  in  the  corollary  to  Art.  140,  this  is 
merely  a  special  form  of  the  condition 

H2  —  AB<0. 

Hence,  the  Circle  is  a  particular  case  of  the  Ellipse. 
Resuming,  then,  the  equation  to  the  Ellipse,  namely, 

(a2  —  c2)  x2  +  a2?/2  =  a2  (a2  —  c2), 

we  notice  that  it  already  fulfills  the  condition  ff=Q. 
Adding  the  condition  A  =  B,  necessary  to  make  it  rep- 
resent a  circle,  we  obtain,  as  characteristic  of  the  Circle, 

a2  —  G2  =  a2  .-.  c  =  Q. 

We  hence  learn  that  the  Circle  is  an  ellipse  whose  two 
foci  have  become  coincident  at  the  center. 

Moreover,  the  Circle  is  real,  vanishes,  or  is  imaginary, 
on  the  same  conditions  as  the  Ellipse.  For  we  saw 
(Art.  142,  Cor.  2)  that  it  assumes  these  several  phases 
according  as  the  quantity 


is  positive,  zero,  or  negative.  Now,  if  we  apply  to  the 
Discriminant  A  the  conditions  H=  0,  A  =  B,  we  find, 
as  true  for  the  Circle, 

-A  =  A(G*  +  F*  —  AC). 

And  since  we  are  always  to  suppose  A  positive,  we  have 
A  <  0  .*.  a  real  circle. 
A  =  0  .•.  a  point. 
A  >  0  .*.  an  imaginary  circle. 

Remark.  —  In  allusion  to  the  fact  that  its  foci  do  not  in  general 
vanish  in  the  center,  the  Ellipse  may  be  called  eccentric. 


THEOREMS  OF  TRANSFORMATION.  163 

1G2.  The  following  table  exhibits  the  analytic  con- 
ditions thus  far  imposed  upon  the  equation  of  the  second 
degree,  with  their  geometric  consequences : 

(  Real     v  A<0. 

II'2  —  AB<0  (ff==±'  A~  B  =  ±  .'.  Eccentric  Curve    .    J  Point  •.•  A=0. 

}  (imag.  •.•  A>0. 

Ellipse        )  (-Real     •.•  A<0. 

^ff=«,A-  B  =  0.  -.Circle J  Point  •.•  A=0. 

(imag.  •.•  A>0. 

163.  Theorems  of  Transformation. — Before  ad- 
vancing further,  it  will  be  well  to  collect  from  the 
foregoing  discussion  the  theorems  it  implies  respecting 
transformation  of  co-ordinates.  They  are  often  conve- 
nient in  performing  the  reductions  to  which  they  relate. 

Theorem  I. — In  transforming  any  equation  of  the  second  degree  to 
parallel  axes  through  a  new  origin: 

1.  The  variable  terms  of  the  second  degree  retain  their  original 
co-efficients. 

2.  The  variable  terms  of  the  first  degree  obtain  new   co-efficients, 
which  are  linear  functions  of  the  new  origin. 

3.  The  constant  term  is  replaced  by  a  new  one,  which  is  the  result 
of  substituting   the   co-ordinates   of  the    new   origin    in   the    original 
equation. 

For,  in  applying  this  transformation  to  equation  (1)  of  Art.  155, 
the  co-efficients  of  x2,  xy,  and  y2  continued  to  be  A,  2H,  and  B ; 
the  co-efficients  of  x  and  y  respectively  replaced  G  and  F  by 

AaS  +  Htf  +  G,   By'  +  Hx'  +  F; 
and,  for  the  new  constant  term,  we  obtained 

Ax/2  +  ZHx'y'  +  Eyn  +  2Gx/  +  2Fy'  +  C. 

Theorem  II. — In  transforming  any  equation  of  the  second  degree 
from  one  set  of  rectangular  axes  to  another,  the  quantities  A  -j-  B, 
H2  —  AB  remain  unaltered. 

For  the  equation  near  the  middle  of  p.  157  is  the  result  of  this 
transformation ;  and  if  we  add  together  the  co-efficients  of  x~  and 
y2  in  it,  after  representing  them  by  A/  and  B' ',  we  obtain 

A/  +  B/  =  A  +  B. 


164  ANALYTIC   GEOMETRY. 

In  likof  manner,  representing  the  co-efficient  of  xy  by  2///,  and 
performing  the  necessary  operations,  we  find 


Theorem  III  —  7/*  in  the  process  of  transforming  an  equation  of  the 
•second  degree,  the  co-efficients  of  x  and  j  vanish,  the  new  origin  is  the 
center  of  the  locus. 

For,  in  that  event,  the  new  equation  will  be  satisfied  equally  by 
x,  y  and  —  x,  —  y  ;  that  is,  all  right  lines  drawn  through  the  new 
origin  to  meet  the  curve  will  be  bisected  by  that  origin. 

Corollary,  —  If  only  one  of  these  co-efficients  vanish,  the  new  origin 
will  lie  on  a  right  line  passing  through  the  center.  For  we  must  then 
have  either 


that  is,  the  co-ordinates  of  the  new  origin  must  satisfy  one  of  the 
equations  (Art.  155)  by  eliminating  between  which  we  determined 
the  center. 

Theorem  IV.  —  If  the  co-efficient  o/xy  vanish,  the  new  reference-axes, 
if  rectangular,  will  be  parallel  to  the  axes  of  the  locus. 

For  when  in  Art.  156*  this  co-efficient  vanished,  with  the  center 
as  origin,  the  new  reference-axes  coincided  with  the  axes  of  the 
locus;  hence,  if  the  origin  is  at  any  other  point,  they  must  be 
parallel  to  them. 


*  As  the  beginner  is  liable  to  misapprehend  the  argument  of  Art.  156, 
it  may  be  well  to  restate  it,  in  the  form  which  the  present  connection 
suggests:  —  When  we  revolved  the  reference-axes  through  the  angle 

9  7-7" 

e  =  y2  tan-  • 


A-B> 

(which  was  found  by  equating  the  co-efficient  of  xy  to  zero}  we  produced  an 
equation  (3)  identical  in  form  with  that  previously  obtained 'for  the 
Ellipse  by  referring  it  to  its  axes.  So  far  then  as  concerned  the  Ellipse, 
the  new  reference-axes  were  identical  with  the  two  lines  which  (Art.  146) 
we  had  described  as  the  axes  of  that  curve.  But  on  account  of  their 
power  to  reduce  the  general  equation  to  a  fixed  form,  these  two  lines  were 
properly  assumed  to  have  a  fixed  relation  to  its  locus,  analogous  to  that 
which  they  bore  to  the  Ellipse  ;  and  hence  were  called  the  "  axes  "  of  that 
locus. 


EXAMPLES  ON  THE  ELLIPSE.  165 

1O4.  These  theorems  not  only  furnish  criteria  for 
selecting  such  transformations  as  will  represent  required 
geometric  conditions,  but  they  enable  us  to  shorten  the 
process  of  transformation. 

Thus,  knowing  Theorem  I,  we  can  henceforth  write 
the  result  of  transforming  to  parallel  axes,  without  going 
through  with  the  ordinary  substitutions. 

Knowing  Theorem  II,  we  can  immediately  write  the 
central  equation  of  a  second  order  curve  from  its  general 
equation,  by  merely  setting  down  the  first  three  given 
terms  and  adding  a  new  constant  term  found  as  in 
Theorem  I,  3. 

By  uniting  Theorems  II  and  IV,  we  may  shorten  the 
process  of  reduction  to  the  axes.  For,  if  such  a  reduc- 
tion is  required,  we  shall  have  Hf  =  0  ;  and,  therefore, 
A'  +  &=A  +  B  with  A'B'  =  AB  —  H2:  two  equa- 
tions from  which  we  can  easily  find  A'  and  Br.  Cr  is 
found  as  in  Theorem  I,  3.  It  is  preferable,  however,  to 
write  the  reduced  equation  at  once  ;  for  its  form  is 
A'x2  +  B'y2  +  C'  =  Q:  in  which  A',  Br  are  found  by 
formulae  (d)  and  (e)  Art.  156,  and  Cr  is  obtained  as 
before.  When  the  given  equation  is  already  central, 
this  reduction  becomes  very  brief;  since  we  do  not  then 
have  to  calculate  C'. 

EXAMPLES. 

I.    NOTATION    AND    CONDITIONS. 

1.  Determine  by  inspection  the  locus  of  each  of  the  equations 
2x*  +  3?/2  =  12, 


2.  Transform  3z2  +  4.ry  +  if  —  5x  —  by  —  3  =  0  to  parallel  axes 
through  (2,  3).     Is  the  curve  an  ellipse? 


166  ANALYTIC  GEOMETRY. 

3.  Reduce  x*+  2xy— y1  +  Sx  +  4y  —  8  =  0  to  the  center.    Does 
this  represent  an  ellipse  ? 

4.  If  in  a  given  equation  of  the  second  degree  //  —  0,  what 
condition  must  be  fulfilled  in  order  that  the  equation  may  represent 
an  ellipse  ?     What,  if  A  or  _B  equals  zero  ? 

5.  Transform  Ux2  —  4xy  +  lit/2  =  60  to  the  axes  of  the  curve, 
by  all  three  methods.     What  is  the  origin  in  the  given  equation? 
What  is  the  locus,  and  is  the  same  locus  indicated  by  the  reduced 
equation  ? 

6.  Find    the    center    of    5z2  +  4xy  -f  if  —  5x  —  2y  —  19,    and 
reduce  the  equation  to  it.     Show  that  the  curve  is  a  real  ellipse, 
both  by  the  original  equation  and  the  reduced  one. 

7.  Show  that  14-r2  —  4xy  -f-  lly2  =  0  denotes  an  infinitely  small 
ellipse ;  that  is,  an  ellipse  in  the  limiting  case. 

8.  Show  that  5;c2  +  4zy  -f  if  —  5x  —  2y  -f  19  =  0  represents  an 
imaginary  ellipse,  and  verify  by  writing  the  equation  as  referred  to 
the  axes. 

9.  Find  the  equation  to  the  Ellipse,  the  origin  3/y'  being  on  the 
curve,  and  the  reference-axes  parallel  to  the  axes  of  the  curve. 

10.  Find  the  equation  to  an  ellipse  whose  conjugate  axis  is  equal 
to  the  distance  between  its  foci,  taking  for  axes  the  two  lines  that 
join  the  extremities  of  the  conjugate  to  the  left-hand  focus. 

II.    ELLIPTIC  LOCI. 

1.  Find  the  locus  of  the  vertex  of  a  triangle,  given  the  base 
and  the  product  of  the  tangents  of  the  base  angles. 

Taking  the  base  and  a  perpendicular  through 
its  middle  point  for  axes,  calling  the  given  product 
I*  :  «2,  and  in  other  respects  retaining  the  notation 
of  Ex.  1,  p.  126,  we  shall  have 


Q 


7/ 

Hence,  by  the  conditions  of  the  problem,  m-/_  ^  =  —  2  J  anc*  ^Q  equation 
to  the  required  locus  is 


Therefore,  (Art.  147)  the  vertex  moves  on  an  ellipse  whose  center  is  the 
middle  point  of  the  base,  and  whose  foci  are  on  the  base  at  a  distance  from 
the  center  —  m)/n2  —  I2  :  n. 


ELLIPTIC  LOCI. 


167 


2.  Find  the  locus  of  the  vertex,  when  the  base  and  the  sum 
of  the  sides  are  given. 

3  Find  the  locus  of  the  vertex,  given  the  base  and  the  ratio 
of  the  sides. 

4.  Given  the  base,  and  the  product  of  the  tangents  of  the  halves 
of  the  base  angles :  to  find  the  locus  of  the  vertex. 

5.  Two  vertices  of  a  given  triangle  move  along  two  fixed  lines 
which  are  at  right  angles :  to  find  the  locus  of  the  third. 

6.  A  right  line  of  given  length  moves  so  that   its   extremities 
always  lie*  one  on  each  of  two  fixed  lines  at  right  angles  to  each 
other:   to  find   the  locus  of  a  point  which  divides  it  in  a  given 
ratio. 

7.  In  a  triangle  of  constant  base,  the  two  lines  drawn  through 
the  vertex  at  right  angles  to  the  sides  make  a  constant  intercept 
on  the  line  of  the  base :  find  the  locus  of  the  vertex. 

8.  The  ordinate  of  any  circle  a2  +  /32  =  r2  is  moved  about  its 
foot  so  as  to  make  an  oblique  angle  with  the  corresponding  diam- 
eter: find  the  locus  of  its  extremity  in  its  new  position. 

9.  The  ordinate  of  any  circle  x2  -f-  y1  =  r*  is  augmented  by  a 
line  equal  in  length  to  the  corresponding  abscissa:   find  the  locus 
of  the  point  thus  reached. 

10.  To  the  ordinate  of  any  circle  there  is  drawn  a  line,  from 
the  vertex  of  the  corresponding  diameter,  equal  in  length  to  the 
ordinate:  find  the  locus  of  the  point  of  meeting. 

1 1 .  In  any  ellipse,  find  the  locus  of  the  middle  point  of  a  focal 
radius. 

12.  Find  the  locus  of  the  extremity  of  an  elliptic  radius  vector 
prolonged  in  a  constant  ratio. 

13.  A  right  line  is  drawn  through  a  fixed  point  to  meet  an  ellipse : 
find  the  locus  of  the  middle  point  of  the  portion  intercepted  by  the 
curve. 

14.  Through  the  focus  of  an  ellipse,  a  line  is  drawn,  bisecting 
the  vectorial  angle,  and  its  length  is  a  geometric  mean  of  the  radius 
vector  and  the  distance  from  the  focus  to  the  center;  find  the  locus 
of  its  extremity. 

15.  Through  any  point  Q  of  an  ellipse,  a  line  is  drawn  parallel 
to  the  transverse  axis,  and  upon  it  QP  is  taken  equal  to  the  corre- 
sponding focal  radius :  find  the  locus  of  P. 


1G8  ANALYTIC  GEOMETRY. 

SECTION  VI.  —  THE  HYPERBOLA. 

I.    GEOMETRIC    POINT    OF   VIEW: THE     EQUATION    TO     THE 

HYPERBOLA   IS    OF    THE    SECOND    DEGREE. 

165.  The  Hyperbola  is  characterized  by  the  following 
property  :  The  difference  of  the  distances  from  the  variable 
point  of  the  curve  to  two  fixed  points  is  constant. 

Hence,  we  may  trace  the  curve,  and  determine  its  figure, 
as  follows  :  —  Take  any  two  points,  as  F'  and  F.  At  F', 
pivot  the  corner  of  a  ruler 
F'R;  at  F,  fasten  one  end 
of  a  thread,  whose  length  is 
less  than  that  of  the  ruler. 
Then,  having  attached  the 
other  end  to  the  ruler  at  R, 
stretch  the  thread  close  against  the  edge  of  the  ruler 
with  the  point  of  a  pencil  P.  Move  the  ruler  on  its 
pivot,  and  slide  the  pencil  along  its  edge  so  as  to  keep 
the  thread  continually  stretched  :  the  path  of  the  pencil- 
point  will  be  an  hyperbola.  For,  in  every  position  of  _P, 
we  shall  have 

F'P—  FP=  (F'P  +  PR)  —  (FP  +  PR)  =  F'R—FPR. 

That  is,  the  difference  of  the  distances  from  the  variable 
point  P  to  the  two  fixed  points  F'  and  F  will  always  be 
equal  to  the  difference  between  the  fixed  lengths  of  the 
ruler  and  the  thread ;  or,  we  shall  have 

F'P  —  FP  =  constant. 

By  pivoting  the  ruler  at  jP,  and  fastening  the  thread 
at  F',  we  shall  obtain  a  second  figure  similar  in  all 
respects  to  the  former,  except  that  it  will  face  in  the 
opposite  direction.  The  complete  curve  therefore  con- 
sists of  two  branches,  as  represented  in  the  diagram. 


EQUATION  TO  HYPERBOLA.  169 


166.  The  two  fixed  points,  Ff  and  F,  are  called  the 
foci  of  the  curve  ;  and  the  variable  distances  with  a  con- 
stant difference,  F'P  and  FP,  are  termed  its  focal  radii. 

The  portion  A'  A  of  the  right 
line  drawn  through  the  foci,  is 
called  the  transverse  axis.  The 
point  0,  taken  midway  between 
the  foci,  we  shall  for  the  present 
call  the  focal  center. 

It  is  apparent  from  the  diagram,  that  the  right  line 
YfY,  drawn  through  the  focal  center  at  right  angles  to 
the  transverse  a.xis,  does  not  meet  the  curve.  We  shall 
find,  however,  that  a  certain  portion  of  it  has  a  very 
significant  relation  to  the  Hyperbola,  and  is  convention- 
ally known  as  the  conjugate  axis.  For  the  present,  we 
shall  speak  of  the  whole  line  under  that  name. 

167.  Equation  to  the  Hyperbola,  referred  to  its 
Axes.  —  Putting   2<?   =   the    distance   between   the    foci, 
and  2a  ==  the  constant  difference  of  the  focal  radii,  we 
shall  have  from  the  diagram  above,  F  '  P2  =  (x  -\-  c)2  +  y2 
and  FP2  =  (x  —  cf  -f-  y2.     The  defining  property  of  the 
Hyperbola  will  therefore  be  expressed  by 

V{(x  4  c}2  +  y2}  -  i/{(x  -  c)2  +  y2}  =  2a. 
Clearing  of  radicals,  we  obtain 

(c2  —  a2)  x2  —  a2y2  =  a2  (c2  —  a2)  (1)  ; 

and,  by  writing  b2  for  c2  —  a2  in  order  to  abbreviate,  the 
required  equation  becomes 

b2x2  —  a2)/2  =  a2b2  (2)  : 

which,  on  the  analogy  of  the  equations  to  the  Right  Line 
and  the  Ellipse,  may  be  written 


a 


170  ANALYTIC  GEOMETRY. 

Corollary. — Hence  we  may  regard  the  b2  of  the 
Hyperbola  as  the  negative  *  of  the  b2  of  the  Ellipse,  and 
we  infer  the  following  principle  :  Any  function  of  b  that 
expresses  a  property  of  the  Ellipse,  will  be  converted  into 
one  expressing  a  corresponding  property  of  the  Hyperbola 
by  merely  replacing  its  b  %  b  I/ — 1. 

Remark. — The  relation  thus  suggested  between  these  two  curves 
will  display  itself  completely  when  we  come  to  discuss  their  prop- 
erties. Results  will  continually  occur,  which  give  color  to  the 
fancy  that  an  hyperbola  is  a  reversed  ellipse. 

16S.  We  must  next,  as  in  the  case  of  the  Ellipse, 
investigate  the  general  form  of  the  equation  we  have 
obtained. 

Taking  it  up  in  the  form  (2),  namely, 

b2x2  —  a2y2  =  a2b2, 

we  speedily  discover  that  a2  and  b2  must  have  like  signs ; 
for  if  their  signs  were  unlike,  the  equation  would  assume 
one  or  the  other  of  the  forms 

b2x2  +  ay  —  a2b2  =  0, 
b2x2  +  ahf  -f  a2b2  =  0, 

and  thus  would  denote  (Art.  148)  not  an  hyperbola,  but 
an  ellipse.  Hence,  in  the  equation  to  the  Hyperbola 
referred  to  its  axes,  the  co-efficients  of  x2  and  y2  must 
have  unlike  signs. 

This  condition  may  be  fulfilled  either  by  supposing 
a2  and  b2  both  positive  or  both  negative.  On  the  former 
supposition,  the  equation  will  be 

b2x2  —  a2y2  —  a2b2  =  0  ; 


#  It  must  be  remembered  that  I2  is  only  an  abbreviation.  All  that  is 
meant,  then,  by  the  expression  in  the  test  is,  that  the  operation  for 
which  b2  stands  in  the  Hyperbola  is  the  reverse  of  the  corresponding  one 
in  the  Ellipse. 


ITS  GENERAL  FORM. 


171 


and    on    the    latter,    after    changing   its    signs,   it   will 
become 

6V  —  a2y2+a2b2  =  0. 

What,  now,  is  the  geometric  meaning  of  the  variation 
in  the  sign  of  a252,  which  is  thus  brought  into  view  ?  In 
the  case  of  the  Ellipse  it  indicated  (Art.  148)  the  transi- 
tion from  the  real  to  the  imaginary  state  of  the  curve  ; 
and  (as  the  transition  to  -j-  a2b2  was  made  by  replacing 
a  and  b  by  a  V  —  1  and  b  V  —  1)  we  might  suppose  that 
it  indicated  the  same  thing  here,  were  it  not  that  a  dif- 
ferent conclusion  is  rendered  certain  by  the  following 
considerations. 

Let  us  conceive  of  an  hyperbola  whose  foci  Ff  and  F 
are  at  the  same  distance  from  their  center  0  as  those  of 
the  curve  already  considered,  but  lie  upon  the  conjugate 
axis  instead  of  the  transverse. 
Then,  retaining  the  same  axes  of 
reference  as  before,  we  shall  evi- 
dently have,  for  the  new  positions 
of  F'  and  F, 


Supposing  now  that,  in  addition,  the  constant  difference 
of  the  focal  radii  in  the  new  curve  is  26  instead  of  2a, 
its  equation  will  be 

!/{z2  +  (c  +  </)2}  -V{x2+  (c  —  y)2}  =  26; 
or,  after  clearing  of  radicals, 

(c2  —  b2)  y2  —  b2x2  =  b2  (c2  —  b2). 
By  substituting  for  c2  —  b2  its  value  a2,  this  becomes 

a2y2  —  b2x2=a2b2, 

and,  by  changing  the  signs  and  transposing, 
b2x2—  a2         a262  =  0: 


172  ANALYTIC  GEOMETRY. 

which  is  precisely  the  expression  we  obtained  above  by 
supposing,  in  the  equation  to  the  original  hyperbola,  both 
a2  and  b2  to  be  negative. 

An  hyperbola  which  thus  has  its  foci  on  the  conjugate 
axis  of  another,  yet  at  the  same  distance  apart,  and 
whose  a  is  the  other's  b,  is  said  to  be  conjugate  to  the 
given  one.  We  learn,  therefore,  that  the  equation  to  the 
Hyperbola  conforms  to  the  general  type 

Ax2  +  B'y2  +  C'  =  0, 

in  which  A'  is  positive,  B'  is  negative,  and  C'  is  negative 
or  positive  according  as  the  curve  is  primary  or  con- 
jugate. 

169.  Theorem, — In  any  hyperbola,  the  focal  center  is 
the  center  of  the  curve. 

For  the  equation  obtained  by  taking  the  focal  center 
as  origin  contains  no  variable  terms  except  such  as  are 
of  the  second  degree.  But  (Art.  163,  Th.  Ill)  the  origin 
for  such  an  equation  is  the  center  of  the  curve. 

170.  Polar  Equation   to   the  Hyperbola,  the 
Center  being  the  Pole. — Changing  b2x2  —  a2y2  =  a2b2  to 
polar  co-ordinates,  we  find  (Art.  57,  Cor.) 

P*  =  b2cos20  —  a2sin20; 
or,  (Trig,  838,) 

azb2 


(a2  +  b2)  cos2  6  —  a2 

Dividing  both  terms  of  the  second  member  by  a2,  and 
putting 


a2 


POLAR  EQUATION  TO  HYPERBOLA.  173 

we    obtain   the   usual   form    of   the   required    equation, 
namely, 


e2  cos2  0  —  1 

Bemark.—  Here,  as  well  as  in  the  case  of  the  Ellipse,  the  equa- 
tion implies  that  for  any  value  of  6  there  are  two  radii  vectores, 
numerically   equal,   with   contrary   signs. 
Here,  too,  the  equation  is  verified  by  the 
diagram;    for  the  two  points,  P  and  Px, 
obviously  correspond  to  the  same  angle  0, 
if  we  fix  the  position  of  P/  by  the  radius 
vector  OP/  =  —OP. 

It  is  also  worthy  of  notice,  that  this  equation  may  be  derived 
from  the  central  polar  equation  to  the  Ellipse  (Art.  150)  by  sub- 
stituting —  6*  for  tf  in  the  latter. 

171.  The  two  abbreviations  employed  above,  namely, 

a2     l     7>2 

c2  —  a2  =  b2  and  -  -33-  =«*, 

a2 

may  evidently  be  derived  from  the  two  used  in  connection 
with  the  Ellipse  (Art.  151),  by  substituting  —  b2  for  b2. 
By  combining  them,  however,  we  still  obtain  the  relation 

c  =ae. 

Corollary,  —  Hence,  the  central  polar  equation  to  the 
Hyperbola  may  be  written 

a2(l-e2)    . 


a  formula  which  we  leave  the  student  to  distinguish 
from  that  given  for  the  Ellipse  in  the  corollary  to 
Art.  151. 


174  ANALYTIC  GEOMETRY. 

172.  Polar   Equation   to   the   Hyperbola,  the 
Focus  being  the  Pole.— From  the  annexed  diagram, 
we  have  FfP  =  p,  and  FP  =  -|/((o2  -f  4c2  —  4pc  cos  0). 
Hence,  expressing  the  defining  prop- 
erty of  the  Hyperbola, 


P  —  I/Go2  +  4c2  —  4fjC  cos  6)=2a: 


c  cos  6  —  a 

Replacing  c  by  its  equal  ae,  we  obtain  the  usual  form 
of  the  equation, 

;        1 — ecosd 

Remark — The  apparent  identity  of  this  equation  with  that  of 
Art.  152,  we  leave  the  student  to  explain;  and  he  may  show  that 
when  the  right-hand  focus  is  the  pole,  the  equation  will  be 

_  a(g'-l) 


P       1  —  e  cos  6 


EXAMPLES. 

1.  Given  the  two  points  (—3,  0)   and   (3,  0);    on  the  first  is 
pivoted  a  ruler  whose  length  =  20,  and  in  the  second  is  fastened 
a  thread  whose  length  =  16:  form  the  equation  to  the  hyperbola 
generated  by  means  of  this  ruler  and  thread  as  in  Art.  165. 

2.  In  a  given  hyperbola,  half  the  difference  of  the  focal  radii 
=  2,  and  half  the  distance  between  the  foci  =  3 :  write  its  equa- 
tion.    Why  can  not  this  example  be  derived  from  Ex.  2,  p.  153,  by 
merely  substituting  "difference"  for  "sum"? 

3.  Form  the  equation  to  the  hyperbola  whose  focus  is  1  foot  from 
its  center,  and  whose  focal  radii  have  the  constant  difference  of  3 
inches. 

4.  In  a  given  hyperbola,  the  difference  of  the  focal  radii  =  8, 
and  the  difference  between  the  squares  of  half  that  difference  and 
half  the  distance  between  the  foci  =  —  9 :  write  its  equation. 


HYPERBOLA  AND  GENERAL  EQUATION.       175 

5.  Write  the  equations  to  the  hyperbolas  which  are  the  conju- 
gates of  those  preceding. 

6.  Write  the  central  polar  equation  to  the  hyperbola  in  which 
the  squares  of  half  the  difference  of  the  focal  radii  and  half  the 
distance  between  the  foci  differ  by  —  9,  and  the  ratio  of  the  dis- 
tance between  the  foci  to  the  difference  of  the  focal  radii  =2:1. 

7.  Write  the  central  polar  equation  to  the  hyperbola  of  Ex.  2. 

8.  Find,  in  the  same  hyperbola,  the  ratio  which  the  difference 
of  the  focal  radii  bears  to  the  distance  between  the  foci,  and  write 
the  equation  in  the  form  given  in  the  corollary  to  Art.  171. 

9.  In  a  given  hyperbola,  the  difference  of  the  focal  radii  =  12, 
and  the  ratio  of  that  difference  to  the  distance  between  the  foci 
=  1:3.     Write  its  polar  equation,  the  focus  being  the  pole. 

10.  The  focus  being  the  pole,  form  the  equation  to  the  hyperbola 
of  Ex.  3.  What  would  this  be,  if  the  right-hand  focus  were  the 
pole? 

II.    ANALYTIC    POINT    OF   VIEW:  —  THE    EQUATION    OF    THE 

SECOND    DEGREE    ON   A    DETERMINATE    CONDITION 

REPRESENTS    AN   HYPERBOLA. 

173.  To  establish  this  theorem,  we  must  show  (Art. 
168)  that  the  general  equation  of  the  second  degree  is 
reducible  to  the  form 


in  which  A!  is  positive,  and  Br  negative  ;  and  we  must 
be  able  to  find  real  conditions  upon  which  the  reduction 
can  always  be  effected. 

From  (3)  of  Art.  156,  we  already  know  that,  apart 
from  the  question  of  signs^  the  general  equation  is  re- 
ducible to  the  required  form.  It  only  remains,  then,  to 
determine  the  condition  which  must  be  fulfilled  in  order 
that  the  signs  of  A'  and  B'  may  be  such  as  characterize 
the  Hyperbola. 
An.  Ge.  18. 


176 


ANA  L  YTIC   G  EG  METE  Y. 


174.  If  A'  is  positive,  and  B'  negative,  the  product 
A'B'  must  be  negative.     Therefore   (see   Art.  158)  for 
the  Hyperbola  we  shall  have 

AB  —  H2  =  negative  ; 
or,  after  changing  the  signs  throughout, 
H2  —  AB  =  positive. 

Hence,  The  equation  of  the  second  degree  represents  an 
hyperbola  whenever  its  co-efficients  fulfill  the  condition 

H2  -AB>0. 

175.  Let  us  now  inquire  what  additional  criteria  the 
equation  must  satisfy,  in  order  to  distinguish  between  a 
primary  hyperbola  and  its  conjugate. 

For  the  reduced  form  of  the  equation,  (Art.  168,)  the 
curve  is  primary  or  conjugate  according  as  C'  is  nega- 
tive or  positive.  But  [Art.  155,  (b)]  we  have 


J 


C'=  — 


in  the  case  of  the  Hyperbola  therefore,  since  H2  —  AB 
is  positive,  C1  will  be  negative  when  J  is  positive,  and 
positive  when  J  is  negative.  Hence, 

H2  -  AB  >  0  with  J  >  0 

is  the  condition  that  the  equation  of  the  second  degree 
shall  represent  a  primary  hyperbola;  and 

H2  —  AB  >  0  with  J  <  0 
is  the  condition  that  it  shall  represent  a  conjugate  one. 


RIGHT  LINE  AS  HYPERBOLA. 


177 


17G.   The  limit   separating   the   two   conditions   just 
determined,  is  evidently 

0  with  J  =  0. 


But  (Arts.  131,  132;  cf.  160)  this  is  the  condition  that 
the  equation  of  the  second  degree  shall  represent  two 
real  right  lines  intersecting  in  the  center  of  its  locus. 
Hence,  Two  real  right  lines,  intersecting  in  its  center, 
form  the  limiting  case  of  the  Hyperbola. 

Kemark  —  The  student  may  verify  this  by  applying  the  condition 
A  =  0  to  the  equation  A'x*  -  B'f  +  C'  =  0. 

177.  In  Art.  161,  we  found  the  Circle  to  be  a  partic- 
ular case  of  the  Ellipse.  We  shall  now  see  that  the 
Hyperbola  has  an  analogous  case. 

One  way  of  satisfying  the  criterion  of  the  Hyperbola 
is,  to  have  in  the  general  equation  the  condition 

A  +  £  =  (). 

For  then  H2  —  AB  will  become  H2  +  A2  :  which  is 
necessarily  positive.  But  if  A  -f-  B  =  0,  then  (Art.  163, 
Th.  II)  A'  +  B'  =  Q;  and,  after  dividing  through  by  A', 
the  equation  referred  to  the  axes  will  become 

x2  —  y2  —  constant  : 

an  expression  denoting  an  hyperbola,  by  the  condition 
just  established,  and  strongly  resembling  the  equation  to 
the  Circle, 

x2  -f-  if  =  constant. 

Corollary.  —  Suppose  now  we  push  this  hyperbola  to  its 
limiting  case.  Its  equation  will  of  course  continue  to 
fulfill  the  condition  A  -}-  B  =  Q,  and  the  corresponding 
pair  of  right  lines  will  therefore  (Art.  128,  Cor.)  intersect 
at  right  angles.  Accordingly,  this  curve  is  known  as  the 
Rectangular  Hyperbola. 


178  ANALYTIC  GEOMETRY. 

178.  The  following  table  presents,  in  their  proper 
subordination,  the  several  conditions  by  which  we  may 
distinguish  the  varieties  of  the  Hyperbola  in  the  equation 
of  the  second  degree  : 

(  Primary  •.*  A  >  0. 

'  A  +  B  =  ±  .'.  Oblique  .  .    j  Two  Intersect.  Lines  •.•  A=0. 
H*-AB>Q  '•Conjugate  •.•  A  <  0. 

Hyperbola  /-Primary  -.-  A  >  0. 

A  +  B=Q  .'.   Rectangular  -j  Two  Perpendiculars  •.•  A  =  0. 
'-Conjugate  •.•  A<0. 

By  comparing  this  table  with  that  of  Art.  162,  the 
student  will  see  the  truth  of  the  Remark  under  Art.  167. 


EXAMPLES. 
I.    NOTATION   AND   CONDITIONS. 

1.  Determine  whether  the  following  equations  represent  hyper- 
bolas : 

3z2  —   Sxy  +  5?/2  —  6z  +    4y  —  2  =  0, 

*'+    2xy—   y'2  +    x-    3*/  +  7  =  0, 
5z2  —  \2.xy  —  7y2  +  Sx  —  IQy  +  3  =  0. 

2.  Show  that  2z2  —  I2xy  +  5y2  —  6x  +  8y  —  9  =  0  represents  an 
oblique  primary  hyperbola. 

3.  Show  that  3z2  +  8xy  —  3y--\-6x—lQy  +  5  =  Q  represents  a 
rectangular  conjugate  hyperbola. 

4.  Show  that  3z2  -f  Sxy  —  3y2  +  6x  +  lOy  —  5  =  0  represents  a 
rectangular  primary  hyperbola. 

5.  Verify  the  proposition  of  Ex.  2  by  reducing  the  equation  to 
the  axes  of  the  curve. 

6.  Verify  the  propositions  of  Ex.  3  and  4  in  the  same  manner. 

7.  Given  the  hyperbola  5o;2  —  6y2  =  30 :  form  the  equation  to  its 
conjugate,  and  find  the  quantities  a,  6,  c,  and  e. 

8.  Show   that   2x2  +  xy  —  1 5?/2  —  x  +  19y  —  6  =  0   denotes   an 
oblique  hyperbola  in  its  limiting  case,  and  find  the  corresponding 
center. 


HYPERBOLIC  LOCI. 


179 


9.   Show  that  3*2  —  Sxy  —  3?/2  -f  x  +  Yly  —  10  =  0   denotes    a 
rectangular  hyperbola  in  its  limiting  case,  and  find  the  center. 

10.  Transform  the  two  equations  last  given,  to  the  centers  of  their 
respective  curves. 

II.   HYPERBOLIC   LOCI. 

1.  Given  the  base  of  a  triangle,  and  the  difference  of  the  angles 
at  the  base :  to  find  the  locus  of  the  vertex. 

Since  the  difference  of  the  base  angles  is 
given,  the  tangent  of  their  difference  is  given. 
Let  us  call  it  =  -^  .  Then,  using  the  axes  and 
notation  of  Ex.  1,  p.  126,  we  shall  have 


II 


m  —  x       m  +  x 


1  H 5^         5 


a. 

h'>    or> 


m*  -  x*+y*      h 


Hence,  the  equation  to  the  required  locus  is 

ax2  +  2hxy  —  a.yi  =  am2 ; 

and  the  vertex  moves  (Art.  177)  on  a  rectangular  hyperbola  whose  center 

[Art.  155,  (2)]   is  the  middle  point  of  the  base,  and  whose  transverse  axis 

h 
[Art.  156,  (c)]  is  inclined  to  the  base  at  an  angle  0  =  %  tan""1  —  ;  that  is,  at  an 

angle  =  half  the  complement  of  the  given  difference  between  the  base  angles. 

2.  Given  the  base  of  a  triangle,  and  the  difference  between  the 
tangents  of  the  base  angles :  to  find  the  locus  of  the  vertex. 

3.  Find  the  locus  of  the  vertex,  given  the  base  of  a  triangle, 
and  that  one  base  angle  is  double  the  other. 

4.  Find  the  locus  of  the  vertex  in  an  isosceles  triangle,  when 
the  extremity  of  one  equal  side  is  fixed,  and  the  other  equal  side 
passes  through  a  fixed  point. 

5.  Given  the  vertical  angle  of  a  triangle,  and  also  its  area: 
find  the  locus  of  the  point  where  the  base  is  cut  in  a  given  ratio. 

6.  Find  the  locus  of  a  point  so  situated  in  a  given  angle,  that, 
if  perpendiculars  be  dropped  from  it  upon  the  sides  of  the  angle, 
the  quadrilateral  thus  formed  will  be  of  constant  area. 


180 


ANALYTIC  GEOMETRY. 


M     Q,  X 


7.  Given  a  fixed  point  and  a  fixed  right  line :  to  find  the  locus 
of  P,  from  which  if  there  be  drawn  a  right  line  to  the  fixed  point 
and  a  perpendicular  to  the  fixed  line,  they  will  make  a  constant 
intercept  on  the  latter. 

8.  In  the  annexed  diagram,  QP  is  perpen- 
dicular to  OQ,  and  RP  to  072:  find  the  locus 
of  P,  on  the  supposition  that  QR  is  constant. 

9.  Supposing  that  QR  in  the  same  diagram 
passes  through  a  fixed  point,  find  the  locus  of 
the  intersection  of  two  lines  drawn  through  Q 
and  R  parallel  respectively  to  OR  and  O  Q. 

10.  QR  is  a  line  of  variable  length,  revolving  upon  the  fixed 
point  a/3:  find  the  locus  of  the  center  of  the  circle  described  about 
the  triangle  ORQ. 

11.  QR  moves  between  OQ  and  072  so  that  the  area  of  the 
triangle  ORQ   is  constant:    find   the  locus  of  the  center  of  the 
circumscribed  circle. 

12.  A  circle  cuts  a  constant  chord  from  each  of  two  intersecting 
right  lines :  find  the  locus  of  its  center. 

13.  Find  the  locus  of  the  middle  point  of  any  hyperbolic  focal 
radius. 

14.  From  the  extremity  of  any  hyperbolic  focal  radius  a  line  is 
drawn,  parallel  to  the  transverse  axis  and  equal  in  length  to  the 
radius :  find  the  locus  of  its  extremity. 

15.  The  ordinate  of  an  hyperbola  is  prolonged  so  as  to  equal  the 
corresponding  focal  radius:  find  the  locus  of  the  extremity  of  the 
prolongation. 


SECTION  VII.  —  THE  PARABOLA. 


I.    GEOMETRIC    POINT    OF   VIEW:  —  THE    EQUATION   TO    THE 
PARABOLA   IS    OF   THE    SECOND   DEGREE. 

1TO.  The  Parabola  may  be  defined  by  the  following 
property:  The  distance  of  the  variable  point  of  the  curve 
from  a  fixed  point  is  equal  to  its  distance  from  a  fixed 
right  line. 


EQUATION  TO  PARABOLA. 


181 


We  may  therefore  trace  the  curve  and  find  its  figure 
as  follows :  —  Take  any  point  F,  and  draw  any  right  line 
D'D.  Along  the  latter,  fix  the  edge  of  a  ruler;  and  in 
the  former,  fasten  one  end  of  a  thread  whose  length  is 
equal  to  that  of  a  second  ruler  RD,  which  is  right-angled 
at  D.  Then,  having  attached  the  other 
end  to  this  ruler  at  JR,  keep  the  thread 
stretched  against  the  edge  RD  with  the 
point  P  of  a  pencil,  while  the  ruler  is 
slid  on  its  edge  QD  along  D'D  toward  F: 
the  path  of  P  will  be  a  parabola.  For, 
in  every  position  of  P,  we  shall  have 

FP=  PD, 

as  these  distances  will  in  all  cases  be  formed  by  subtract- 
ing the  same  length  RP  from  the  equal  lengths  of  the 
thread  and  ruler. 

18©.  The  fixed  point  F  is  called  the  focus  of  the  par- 
abola, and  the  fixed  line  D'D  its  directrix. 

The  line  OF,  drawn  through  the 
focus  at  right  angles  to  the  directrix, 
and  extending  to  infinity,  is  called  the 
axis  of  the  curve.  The  point  A, 
where  the  axis  cuts  the  curve,  is 
termed  the  vertex. 

We  shall  refer  to  the  distance  FP  under  the  name 
of  the  focal  radius. 

181.  Equation  to  the  Parabola,  referred  to  its 
Axis  and  Directrix. — By  putting  2p  =  the  constant 
distance  of  the  focus  from  the  directrix,  we  shall  have, 
in  the  above  diagram,  FP=  ~]/{(x  —  2p)2  -f  y*}  and 
PD  =  x.  The  algebraic  expression  for  the  defining 
property  of  the  Parabola  will  therefore  be 


182 


ANALYTIC  GEOMETRY. 


Clearing  of  radicals,  and  reducing,  we  find  the  required 
equation, 

y*  =  ±p(x  —  p). 


.  Let  us  now  investigate  the  general  type  to  which 
this  equation  conforms.     It  may  evidently  be  written 

f-±px  +  4p2  =  0  (1), 

and  is  therefore  a  particular  case  of  the  general  equation 

+<7'  =  0  (2), 


in  which  B',  Gr,  Cr  are  any  three  constants  whatever. 
Accordingly  our  real  object  is,  to  determine  whether 
every  equation  in  the  type  of  (2)  represents  a  parabola. 
We  may  settle  this  point  as  follows  : 

Let  us  transform  (1)  to  parallel  axes  whose  origin  is 
somewhere  on  the  primitive  axis  of  z,  say  at  the  distance 
xf  from  the  given  origin.  To  effect  this,  we  merely  re- 
place the  x  of  (1)  by  x'  -f-  x,  and  thus  obtain 


y2  —  4px 


—  x'}  —  0 


(3). 


Now,  since  (1)  represents  a  parabola,  (3)  also  does. 
But  in  (3),  since  x'  is  arbitrary,  the  absolute  term  may 
have  any  ratio  whatever  to  the  co-efficient  of  x.  More- 
over, by  taking  x'  of  the  proper  value,  we  can  render  the 
absolute  term  positive  or  negative  at  pleasure,  and,  by 
supposing  2j9  susceptible  of  the  double  sign,  we  shall 
accomplish  the  same  with  respect  to  the  co-efficient  of  x. 
By  carrying  out  these  suppositions,  and  then  multiplying 
the  whole  equation  by  some  arbitrary  constant,  we  can 
give  it  three  co-efficients  which  will  be  entirely  arbitrary, 
and  may  therefore  write  it 


POLAR  EQUATION  TO  PARABOLA. 


183 


To  show,  then,  that  every  equation  in  the  type  of  (2) 
represents  a  parabola,  we  have  only  to  prove  that  2p 
may  be  either  positive  or  negative,  without  affecting  the 
form  of  the  curve  represented  by  (1). 

Now  this  supposition  is  clearly  correct  ;  for  (see  dia- 
gram, Art.  179)  a  negative  value  of  2p  merely  indicates 
that  the  focus  is  taken  on  the  left  of  D'D,  instead  of  on 
the  right:  while,  by  using  the  thread  and  ruler  on  the 
left  of  the  directrix,  we  can  certainly  describe  a  curve 
similar  in  all  respects  to  PAL,  except  that  it  will  face 
in  the  opposite  direction. 

Hence  we  conclude  that  the  equation  to  the  Parabola 
may  always  be  written  in  the  form 

5y  +  2G'x+C'  =  0, 

J9',  6rr,  Cf  being  any  three  constants  whatever  ;  and,  con- 
versely, that  every  equation  of  this  form  represents  a 
parabola. 

183.  Polar  Equation  to  the  Parabola,  the 
Focus  being  the  Pole.  —  From  the  annexed  diagram, 
we  have  FP  =  p  and  OM=  OF  +  FM=2p  +  p  cos  0. 
Accordingly,  the  polar  expression  for  the 
fundamental  property  of  the  Parabola 
will  be 

p  =  2p  -+-  p  cos  0.  o 

The  required  equation  is  therefore 


—  costf" 

Hemark  —  This  expression  implies  that  the  angle  0  is  measured 
from  FX  toward  the  left.  The  student  may  show  that,  supposing  9 
to  be  estimated  from  FO  toward  the  right,  the  equation  to  the 
Parabola  will  be 


P  — 


1+cos 


An.  Ge.  19. 


184  ANALYTIC  GEOMETRY. 

184.  To  exhibit  in  part  the  analogy  of  the  equation 
just  obtained  to  those  of  the  Ellipse  and  Hyperbola  in 
Arts.  152,  172,  let  us  agree  to  write 


as  an  abbreviation  characteristic  of  the  Parabola,  and 
analogous  to  those  adopted  in  Arts.  150,  170  for  the 
other  two  curves.  We  may  then  write  (see  also  Art.  627) 


"    ~ 


1  —  ecosd 


Corollary,  —  Adopting  the  convention  last  suggested, 
we  may  arrange  the  abbreviations  referred  to,  according 
to  their  numerical  order,  thus  : 

Ellipse  .  .  .  .  e  <  1. 
Parabola  .  .  .  .  e  =  l. 
Hyperbola  .  .  .  .  e  >  1. 

EXAMPLES. 

1.  Given  the  points  (4,  0),  (1,  0),  (3,  0):  write  the  rectangular 
equations  to  the  three  parabolas  of  which  they  are  the  foci. 

2.  Write  the  rectangular  equation  to  the  parabola  whose  focus  is 
the  point  (  -  3,  0). 

3.  Transform  the  equations  just  found  to  parallel  axes  passing 
through  the  foci  of  their  respective  curves. 

4.  What  are  the  positions  of  the  foci  with  respect  to  the  direc- 
trices, in  the  parabolas  yl  =  4  (a;  —  4),    4y2  =  —  3  (4x  +  3),    and 
5?/2-6z  -1-9  =  0? 

5.  Write  the  focal  polar  equation  to  the  parabola  whose  focus  is 
2  feet  distant  from  the  directrix,  and  find  the  length  of  its  radius 
vector  when  0  =  90°.    Also,  find  the  polar  equation  to  any  parabola, 
the  pole  being  at  the  intersection  of  the  axis  and  directrix. 


PARABOLA  AND  GENERAL  EQUATION.        185 

II.    ANALYTIC    POINT    OF   VIEW  :  —  THE    EQUATION    OF    THE 

SECOND  DEGREE  ON  A  DETERMINATE  CONDITION 

REPRESENTS  A  PARABOLA. 

185.  To  establish  this  theorem,  we  must  show  that 
there  are  real  conditions  upon  which  the  general  equation 
of  the  second  degree  may  always  be  reduced  to  the  form 
(Art.  182) 

B'y*  +  2G'x  +  C'  =  0. 

186.  In  the  investigation  on  which  we  are  about  to 
enter,  we  must  confine  our  attention  to  those  equations  of 
the  second  degree  whose  co-efficients  fulfill  the  condition 


For  we  have  already  proved  (Arts.  158,  174)  that 
every  equation  of  the  second  degree  in  which  H2  —  AB 
is  not  equal  to  zero,  represents  either  an  ellipse  or  an 
hyperbola. 

187.  Further  :  The  restriction  just  established  carries 
with  it  the  additional  one,  that,  in  the  equations  we  are 
permitted  to  consider,  the  condition 


can  not  occur. 

For,  reverting  to  the  general  value  of  the  Discrimi- 
nant (Art.  159),  we  have 

J  =  ABC  +  2FGH—  AF2  —  BG2  —  CH2. 

Whence,  multiplying  the  second  member  by  B  :  B, 
adding  H2F2  —  H2F2  to  the  numerator  of  the  result, 
and  factoring,  we  may  write 

_  (H2  -  AB)  (F2  -BC}  —  (HF—BG)2 
J- 


186  ANALYTIC  GEOMETRY, 

But,  by  the  preceding  article,  we  must  assume 

H2  —  AB  =  0  ; 
hence,  for  the  purposes  of  the  present  inquiry, 

(HF—BGY 
A  =  -3- 

Now  the  condition  H2  —  AB  =  Q  obviously  requires 
that  A  and  B  shall  have  like  signs  ;  and  we  have 
agreed  (see  foot-note,  p.  160)  to  write  all  our  equations 
so  that  A  shall  be  positive:  therefore  B,  in  the  present 
inquiry,  is  positive.  Whence  it  follows,  that  the  fore- 
going expression  for  A  is  essentially  negative;  unless 
HF  —  BG  =  0,  when  it  will  vanish.  Our  proposition  is 
therefore  established. 

188.  The  restrictions  of  the  two  preceding  articles 
being  accepted,  our  actual  problem  is,  to  determine 
whether  the  equation 

Ax*  +  ZHxy  +  Bf  +  2Gx  +  2Fy  +0=0      (1), 

in  which  we  suppose  IP  —  AB  —  0,  can  be  reduced  to 
the  form 


that  is,  whether  it  can  be  subjected  to  such  a  transforma- 
tion of  co-ordinates  as  will  destroy  the  co-efficients  of 
x\  xy,  and  y. 

189.  In  the  first  place,  then,  we  can  certainly  destroy 
the  co-efficient  of  xy.  For,  to  effect  this,  we  need  only 
revolve  the  axes  through  an  angle  0,  such  that  [Art.  156, 
(c)~\  we  may  have 


a  condition  compatible  with  any  values  of  A,  H,  and  B. 


PARABOLA  AND  GENERAL  EQUATION.        187 

Making  this  transformation,  therefore,  we  shall  get  an 
equation  of  the  form 

A'x2  -f  B'y2  -f  2G'x  -f  2F'y  -f  C  =  0  : 
in  which  (see  the  third  equation  in  Art.  156) 

A'  =  %  {(A  +  B)  +  (A  —  B)  cos  2d  -f  2H sin  26}, 
Bf  =  J  {(J.  +  B)  —  (A  —  5)  cos  2#  —  2IZ"sin  20}, 
and  (Art.  56,  Cor.  3) 

G'  =  G  cos/?  -f.Fsin0,  F'  =F  cos  6  —  G  sin  6. 

Now,  from   the    value  of  tan  20   above,   (see   foot-note, 
p.  158,)  we  know  that 

sin  2d  =  VTn ms  i  /ouv2i  .      cos  2f)  = 


or,  by  applying  the  condition  H2  =  AB,  and  taking  the 
radical  as  negative,  that 


and  therefore,  by  Trig.,  847,  iv,  and  by  again  applying 
the  condition  H2  =  AB,  that 

H  B 

smd  =  —,frT2_,    TV.  ,    cos  6  =  — rrnrr-jv, 


Substituting  these  values  in  the  expressions  for  Af,  B', 
G',  and  J7',  we  obtain 


so  that  the  proposed  transformation  has  destroyed  the 
co-efficient  of  z2  as  well  as  that  of  xy,  and  (1)  becomes 

B'y*  +  2G'x  +  2F'y  -{-0=0  (2). 


188  ANALYTIC  GEOMETRY. 

BOO.  In  the  second  place,  we  can  certainly  destroy 
the  co-efficient  of  y.  For  our  ability  to  do  so  depends 
on  finding  some  new  origin,  to  which  if  we  transform 
(2)  in  parallel  axes,  the  new  co-efficient  of  y  shall  be 
zero  ;  and  that  we  can  find  such  an  origin  is  easily 
proved.  For,  if  the  new  origin  be  x'y',  the  new  equation 
(Art.  163,  Th.  I)  will  assume  the  form 

B'f  +  2G'x  +  2  (B'y1  +  F1)  y  +  C'  =  0  : 
in  which  the  co-efficient  of  y  will  vanish,  if 


/_       F  _ 

y  —    w  — 


B'  '        VH2  + 


and  y'  being  thus  necessarily  finite  and  real,  while  x'  is 
indeterminate,  there  is  an  infinite  number  of  points, 
lying  on  one  right  line,  to  any  of  which  if  we  reduce 
(2)  by  parallel  transformation,  the  co-efficient  of  y  will 
disappear,  and  (2)  will  become 

_By  +  20'a;+C"  =  0  (3). 

191.  We  see,  then,  that  we  can  reduce  (1)  to  the 
required  form;  and  that,  too,  without  imposing  any 
condition  upon  it  other  than  the  original  one,  that 
H2  —  AB  shall  be  equal  to  zero. 

Hence,  The  equation  of  the  second  degree  will  represent 
a  parabola  whenever  its  co-efficients  fulfill  the  condition 


Corollary.  —  It  follows  from  this,  that  whenever  the 
equation  of  the  second  degree  denotes  a  parabola,  its  first 
three  terms  form  a  perfect  square. 


PARALLELS  AS  PARABOLA.  189 

From  the  restriction  established  in  Art.  187, 
we  conclude  that  the  Parabola  presents  only  two  varieties 
of  the  condition  just  determined.  They  are, 

i.  H2  —  AB  =  0  with  J  <  0. 
ii.  H2  —  AB  =  0  with  J  =  0. 


By  referring  to  Arts.  131,  132,  it  will  be  seen  that  the 
second  of  these  is  identical  with  the  condition  upon 
which  the  equation  of  the  second  degree  represents  two 
parallels. 

Hence,  Two  parallels  constitute  a  particular  case  of  the 
Parabola. 

193.  If  we  apply  the  criterion  of  the  Parabola  to  the 
two  lines 


at  whose  intersection  (Art.  155)  the  center  of  the  second 
order  curve  is  found,  we  shall  have  A  equal  to  the  quotient 
of  H2  by  B;  and  the  two  lines  will  become 


=Q     (n)  : 


which  (Art.  98,  Cor.)  are  evidently  parallel.  Hence, 
since  we  may  always  suppose  that  parallels  intersect  at 
infinity,  the  center  of  a  parabola  is  in  general  situated  at 
infinity. 

194.  To  this  general  law,  however,  the  case  brought 
out  in  Art.  192  presents  a  striking  exception.  For, 
when  the  Parabola  passes  into  two  parallels,  J  vanishes  ; 
and  we  obtain  (Art.  187) 


190  ANALYTIC   GEOMETRY. 

so  that,  in  this  case,  the  lines  in  (n)  become  coincident, 
and  the  center  is  any  point  on  the  line 


We  thus  arrive  at  the  conception  of  the  Right  Line  as 
the  Center  of  Tivo  Parallels. 

Remark  __  The  result  of  this  article  is  fully  corroborated  by  the 
equations  to  the  two  parallels  themselves.     For  (Art.   160)  these 
are  _______ 

Hx  +  By  +  F+  V  F'1  —  BC  =  Q, 


which  obviously  represent  two  lines  equally  distant  from 
Hx+  By  -f-  JF=0. 

195.  Two  parallels,  considered  as  a  variety  of  the 
Parabola,  present  three  subordinate  cases,  each  of  which 
has  its  proper  criterion. 

For,  since  the  equations  to  the  parallels  are 


'1  -  J*C--=  0, 


By  +  F—  VF2  —  BO  =  0, 
we  shall  evidently  have  the  following  series  of  conditions  : 

I.  F2 — BOO  .'.  Two  parallels,  separate  and  real. 
II.  F1 — BO=Q  .'.  Two  coincident  parallels, 
in.  F2 — BC<^0  /.  Two  parallels,  separate  but  imaginary. 

Corollary. — Hence,  The  Right  Line,  as  the  limit  of  two 
parallels,  is  the  limiting  case  of  the  Parabola. 

Remark. — It  is  noticeable  that  the  limit  into  which 
the  two  parallels  vanish  when  F2  -BC=Q,  is  the  line 


EXAMPLES  ON  THE  PARABOLA. 


191 


which  we  have  just  shown  to  be  the  center  of  the  rec- 
tilinear case  of  the  Parabola. 

196.  The  results  of  the  foregoing  articles,  as  fixing 
the  varieties  of  the  Parabola  and  their  corresponding 
analytic  conditions,  may  be  summed  up  in  the  following 
table : 

H*—AB=Q     A<0-'-  Center  atlnfinity. 


Parabola. 


{Two  Real  Parallels  •.•  F2—  BC>Q. 
Single  Eight  Line  v  F*-BC=Q. 
Twolmag.  Parallels  '.' 


EXAMPLES. 
I.    NOTATION   AND   CONDITIONS. 

1.  Show  why  the  following  equations  represent  parabolas: 

4x*  -f  1  2xy  +  9?/2  +  6x  —  10y  +  5  =  0, 

(2x—  5y)2  =  3a:+4y—  5, 

5?/2  —  Qx  +  1y  —  1  =  0. 

2.  Show  that  the  preceding  equations  represent  true  parabolas, 
having  their  centers  at  infinity  ;  but  that 


denotes  two  parallels,  whose  center  is  the  line 

2x  =  3y. 

3.  Show  that  4x*  —  llxy  +  9y2  +  Sx  —  I2y  -f  5  =  0  denotes  two 
imaginary  parallels,  whose  center  is  the  real  line  2x  —  oy  —  2  —Q', 
and  that  4x2  —  12xy  +  9y2  —  Sa;  +  12y  +  4  =  0  denotes   the  limit 
of  these  parallels. 

4.  Reduce   9z2  —  24xy  +  1  6?/'2  +  4x  —  8y  —  1  =  0  to   the   form 


5.  Show  that  when  a  parabola  breaks  up  into  two  parallels,  the 
line  Hx  -f-  By  +  F=  0  becomes  the  axis  of  the  curve. 


192  ANALYTIC  GEOMETRY. 


II.    PARABOLIC   LOCI. 

1.  Given  the  base  of  a  triangle,  and  the  sum  of  the  tangents 
of  th.3  base  angles :  to  find  the  locus  of  the  vertex. 

Let  us  take  the  base  for  the  axis  of  y,  and  a  per-  / 
pendicular  through  its  middle  point  for  the  axis  of  x. 

Then,  in  the  annexed  diagram,  OM  will  be  the  ordi-  PT 

nate,  and   MP  the  abscissa  of  the  variable  vertex  P.  X' V — 

Therefore,   supposing    the    length  of   the   base  =  2m,  \ 

and  the  given  sum  of  tangents  =  n,  we  shall  have  \ 


m—y 
and  the  equation  to  the  locus  sought  will  be  ny2—  2mx  —  ?im2  =  0. 

Hence,  (see  close  of  Art.  182,)  the  vertex  moves  on  a  parabola  whose  axis 
is  the  perpendicular  through  the  middle  of  the  base. 

By  comparing  this  equation  with  (3)  of  Art.  182,  it  will  be  seen  that 
the  distances  of  the  directrix  and  focus  from  the  base  are  respectively 


2.  Given  the  base  and  altitude  of  a  triangle  :  to  find  the  locus 
of  the   intersection  of  perpendiculars  drawn  from  the  extremities 
of  the  base  to  the  opposite  sides. 

3.  Given  a  fixed  line  parallel  to  the  axis  of  a:,  and  a  movable 
line  passing  through  the  origin:    to  find  the  locus  of  a  point  on 
the  latter,  so  taken  that  its  ordinate  is  always  equal  to  the  portion 
of  the  former  included  between  the  axis  of  y  and  the  moving  line. 

4.  Lines  are  drawn,  through  the  point  where  the  axis  of  a  par- 
abola meets  the  directrix,  so  as  to  intersect  the  curve  in  two  points: 
to  find  the  locus  of  the  points  midway  between  the  intersections. 

5.  Through  any  point  Q  of  a  circle,  OQ  is  drawn  from  the 
center  O,  and  QR  made  a  chord  parallel  to  the  diameter  EOI  and 
bisected  in  S:  to  find  the  locus  of  P,  where  OQ  and  ES  intersect. 

6  Find  the  locus  of  the  center  of  a  circle,  which  passes  through 
a  given  point  and  touches  a  given  right  line.  [Take  given  line  for 
axis  of  y,  and  its  perpendicular  through  given  point  for  axis  of  x.~] 

7.  Given  a  right  line  and  a  circle  :  to  find  the  locus  of  the  center 
of  a  circle  which  touches  both.  [Take  perpendicular  to  given  line, 
through  center  of  given  circle,  for  axis  of  xJ] 


LOCUS  OF  SECOND  ORDER.  193 

S.  OA  is  a  fixed  right  line,  whose  length  =  a ;  about  0,  a  second 
line  POP'  revolves  in  such  a  manner  that  the  product  of  the  areas 
PAO,  P'AO  =  a4,  and  their  quotient  =  cot2  %  POA  :  to  find  the 
locus  of  P  or  P'.  [Polars.] 

9.  Given  the  base  of  a  triangle,  and  that  the  tangent  of  one  base 
angle  is  double  the  cotangent  of  half  the  other:  to  find  the  locus 
of  the  vertex. 

10.  Find  the  locus  of  the  center  of  a  circle  inscribed  in  a  sector 
of  a  given  circle,  one  of  the  bounding  radii  of  the  sector  being  fixed. 


SECTION  VIII.  —  THE   Locus   OF  THE   SECOND 
ORDER  IN   GENERAL. 

197.  We  have  now  seen  that  the  equations  to  the 
Pair  of  Right  Lines,  the  Circle,  the  Ellipse,  the  Hyper- 
bola, and  the  Parabola,  are  all  of  the  second  degree. 
We  have  proved,  too,  that  the  general  equation  of  the 
second  degree  may  be  made  to  represent  either  of  these 
loci,  by  giving  it  co-efficients  which  fulfill  the  proper  con- 
ditions ;  and,  in  the  course  of  the  argument,  it  has  come 
to  light  that  the  Point,  the  Pair  of  Lines  in  their  various 
states  of  intersection,  parallelism,  and  coincidence,  and  the 
Circle,  are  phases  of  the  three  curves  mentioned  last. 

The  latter  result  suggests  the  question,  Is  not  the  gen- 
eral equation,  considered  without  reference  to  any  of  these 
conditions,  the  symbol  of  some  locus  still  more  generic  than 
either  the  Ellipse,  the  Hyperbola,  or  the  Parabola,  of  which 
these  three  curves  are  themselves  successive  phases  ?  It  is 
the  object  of  the  present  section,  to  show  that  this  ques- 
tion, in  a  certain  important  sense,  is  to  be  answered  in 
the  affirmative  ;  and  to  aid  the  student  in  forming  an 
exact  conception  of  what  is  meant  by  the  phrase  Locus 
of  the  Second  Order  in  General. 


194  ^          ANALYTIC  GEOMETRY. 

198.  We  proceed,  then,  to  show  that  such  a  locus 
exists  ;  and  to  explain  the  peculiar  nature  of  the  existence 
which  belongs  to  it. 

In  the  first  place,  a  moment's  reflection  upon  the  dis- 
cussions in  the  preceding  pages  will  convince  us  that 
hitherto  we  have  not  regarded  the  equation 


Ax*  +  ZHxij  +  %2  +  2fe  +  2Fy  +  C=  0        (1) 

in  the  strictly  general  aspect  at  all.  For  we  have  sup- 
posed its  co-efficients  to  be  subject  to  some  one  of  the 
three  conditions 


and  therefore  to  be  actual  numbers,  since  it  is  only  in 
actual  numbers  that  the  existence  of  such  conditions  can 
be  tested.  But,  obviously,  we  can  conceive  of  equation 
(1)  as  not  yet  subjected  to  any  such  conditions,  the 
constants  A,B,C,  F,G,H  not  being  actual  numbers,  but 
symbols  of  possible  ones;  and,  in  fact,  we  must  so  con- 
ceive of  it,  if  we  would  take  it  up  in  pure  generality. 

In  the  second  place,  not  only  does  the  equation  await 
this  purely  general  consideration,  but  when  so  considered 
it  still  has  geometric  meaning.  For,  though  its  co-efficients 
are  indeterminate,  its  exponents  are  numerical  and  fixed  : 
it  therefore  still  holds  its  variables  under  a  constant  law, 
not  so  explicit  as  before,  but  certainly  as  real.  It  is  still 
impossible  to  satisfy  it  by  the  co-ordinates  of  points  taken 
at  random  ;  it  will  accept  only  such  as  will  combine  to 
form  an  equation  of  the  second  degree. 

Since,  then,  we  must  consider  (1)  in  its  purely  general 
aspect  as  well  as  under  special  conditions  ;  since,  even  in 
this  aspect,  it  still  expresses  a  law  of  form;  and  since 
this  law,  consisting  as  it  does  in  the  mere  fact  that  the 


THE  LOCUS  AS  PURE  LA  W.  195 

equation  is  of  the  second  degree,  must  pervade  all  the 
curves  of  the  Second  order:  it  follows  that  this  law 
may  be  regarded  as  a  generic  locus,  whose  properties 
are  shared  alike  by  the  Ellipse,  the  Hyperbola,  and 
the  Parabola. 


By  the  phrase  Locus  of  the  Second  Order  in 
General  we  therefore  mean  not  a  figure  but  an  abstract 
law  of  form.  It  exists  to  abstract  thought,  but  can  not 
be  drawn  or  imagined.  To  illustrate  the  nature  of  its 
existence  by  a  more  familiar  case,  we  may  compare  it  to 
that  of  the  generic  conception  of  a  parallelogram.  We 
can  define  a  parallelogram  ;  but  if  we  attempt  to  imagine 
or  draw  one,  we  invariably  produce  some  particular  phase 
of  the  conception  —  either  a  rhomboid,  a  rhombus,  a  rect- 
angle, or  a  square.  In  the  same  way,  the  Locus  of  the 
Second  Order  exists  so  as  to  be  defined  ;  but  not  other- 
wise, except  in  its  special  phases.  In  short,  when 
speaking  of  it,  we  are  dealing  with  a  purely  analytic 
conception  ;  and  the  beginner  should  avoid  supposing 
that  it  is  any  thing  else. 

2OO.  Further  :  This  common  law  of  form  not  only 
manifests  itself  in  all  the  three  curves  we  have  been 
considering,  but  they  may  be  regarded  as  successive 
phases  of  it,  whose  order  is  predetermined.  For,  as  we 
have  seen,  they  may  be  supposed  to  arise  out  of  the 
general  locus  whenever  the  condition  characteristic  of 
each  is  imposed  on  the  general  equation.  Now  these 
conditions  may  be  summed  up  as  follows  : 


0,  J  =  0,  A  >  0. 


196  ANALYTIC  GEOMETRY. 

Hence,  since  0  lies  between  —  and  +,  while  the  condi- 
tions in  A  actually  entered  our  investigations  as  subordi- 
nates of  those  in  H2  —  AB,  and  the  conditions  in 
F2  —  BC  as  subordinates  of  J  =  0,  it  follows  that  the 
three  curves  have  a  natural  order  corresponding  to  the 
analytic  order  of  their  criteria,  and  that  their  several 
varieties  have  a  similar  order  corresponding  to  theirs. 
Accordingly,  we  should  expect  the  curves  to  occur  thus  : 
Ellipse  ;  Parabola  ;  Hyperbola.  In  due  time  hereafter, 
this  order  will  be  verified  geometrically. 

2O1.  Our  three  curves  and  their  several  varieties  are 
thus  shown  to  be  species  of  the  Locus  of  the  Second 
Order  :  are  there  any  others  ?  We  shall  now  show  that 
there  are  not,  by  proving  that  every  equation  of  the 
second  degree  must  represent  one  of  the  three  curves 
already  considered. 

We  have  just  seen  that  all  the  conditions  hitherto 
imposed  on  the  general  equation  are  subordinate  to  the 
three, 


and  we  have  proved  that  any  equation  of  the  second 
degree  fulfilling  either  of  these  must  represent  one  of  the 
three  curves.  But  no  equation  of  the  second  degree  can 
exist  without  being  subject  to  one  of  these  conditions; 
for,  whatever  be  the  numerical  values  of  A,  .77,  and  B, 
we  can  always  form  the  function  H2  —  AB,  which  can 
not  but  be  less  than,  equal  to,  or  greater  than  zero. 
Hence,  the  series  of  conditions  already  imposed  on  the 
general  equation  exhaust  the  possible  varieties  of  its 
locus,  and  we  have  the  proposition  at  the  head  of  this 
article. 

2O2.  We  mentioned  in  Art.  47,  that  the  term  Conic 
Section    or   Conic  is  used  to  describe   a  curve   of  the 


THE  CONIC  IN  GENERAL.  197 

Second  order.  From  what  has  now  been  shown,  we  may 
define  the  Conic  in  General  as  the  embodiment  of  that 
general  law  of  form  which  is  expressed  by  the  uncondi- 
tioned equation  of  the  second  degree. 

It  also  follows  that  there  are  three  species  of  the  Conic, 
corresponding  to  the  three  leading  conditions  which  have 
become  so  familiar. 

2OB.  Let  us  now  recapitulate  the  argument  by  which 
we  have  thus  gradually  established  the  theorem  :  —  Every 
equation  of  the  second  degree  represents  a  conic. 

I.  We  proved  (Arts.  153  —  162)  that  every  equation  of  the  second 
degree  whose  co-efficients  fulfill  the  condition  H2  —  AB  <  0,  repre- 
sents an  ellipse,  and  showed  that  the  Point,  the  Pair  of  Imaginary 
Lines,  and  the  Circle,  are  particular  cases  of  that  curve. 

II.  We  proved  (Arts.  173  —  178)  that  every  equation  of  the  second 
degree  whose  co-efficients  fulfill  the  condition  H2  —  AB  >  0,  repre- 
sents an  hyperbola,  and  showed  that  the  Pair  of  Real  Intersecting 
Lines  are  a  case  of  that  curve. 

III.  We  proved  (Arts.  185  —  196)  that  every  equation  of  the  second 
degree  whose  co-efficients  fulfill  the  condition  H'L  —  A  B  =  0,  repre- 
sents a  parabola,  and  showed  that  Two  Parallels,  whether  separate, 
coincident,  or  imaginary,  are  a  case  of  that  curve. 

IV.  We  combined  (Art.  200)  the  results  of  the  three  preceding 
steps,  and  inferred  that,  of  the  conditions  previously  imposed,  the  three 

H2  —  AB<0,     H*-AB--=0,     H2 


were  all  that  we  needed  to  consider  in  testing  the  leading  signifi- 
cation of  the  equation  of  the  second  degree,  since  all  the  others 
had  proved  to  be  subordinates  of  these. 

V.  We  showed  (Art.  201)  that  these  three  conditions  are  of  such 
a  nature  that  every  equation  of  the  second  degree  must  be  subject 
to  one  of  them;  and  thence  inferred  the  theorem. 

2O4.  The  existence  of  three  conditions  by  which  the 
signification  of  the  general  equation  may  be  varied,  indi- 
cates, as  we  have  already  noticed,  three  species  of  the 
Conic.  But  we  must  not  overlook  a  previous  subdivision 


198 


ANALYTIC  GEOMETRY. 


of  the  locus.  The  three  conditions  are  themselves  subject 
to  classification :  two  of  them  are  finite,  while  the  third 
is  infinitely  small.  This  classification,  too,  has  its  geo- 
metric counterpart :  for  the  Ellipse  and  Hyperbola,  in 
which  H2  —  AB  is  finite,  have  each  a  finite  point  as  their 
center;  while  the  Parabola,  in  which  H2 -  —  AB  is  infi- 
nitely small,  has  no  such  center.  In  order,  then,  to 
include  all  the  facts,  we  must  say  that  the  Conic  consists 
of  two  families  of  curves,  one  central  and  the  other  non- 
central;  and  that  these  two  families  break  up  into  the 
three  species  which  we  have  already  described. 

2O«5.  The  entire  Locus  of  the  Second  Order,  with  its 
subdivisions  arranged  according  to  their  'mutual  relation- 
ships as  fixed  by  their  analytic  conditions,  may  be  pre- 
sented as  follows: 


ORDER.       FAMILY.       SPECIES. 


CONIC 


Central 


Ellipse 


Hyperbola 


VARIETY. 


Eccentric 


Circle 


Oblique 


CASE. 

Real. 

Point. 

Imaginary. 

Real. 

Point. 

Imaginary. 

Primary. 
Intersect.  Lines. 
Conjugate. 


(Primary. 
Perpendiculars. 
Conjugate. 


f  Center  at  Infinity. 

Non-    \  Parabola  f  Real  Parallels. 

r  -raraooia -j     Center  a       _. 

T>.  I-XT-       \  Single  Line. 
Right  Line      _ 

[  Imag.  Parallels. 


I  Central 


TRILINEAR  CO-ORDINATES. 


199 


CHAPTER   SECOND. 

THE  MODERN  GEOMETRY:   TRILINEAR  AND 
TANGENTIAL  CO-ORDINATES. 

SECTION   I.  —  TRILINEAK   CO-ORDINATES. 

2O6.  Modern  geometers  frequently  employ  tHe  fol- 
lowing method  of  representing  a  point : 

Any  three  right  lines  that  form  a 
triangle,  as  AB,  BC,  CA,  are  assumed 
as  the  Fixed  Limits  to  which  all  posi- 
tions shall  be  referred.     The  position     /A    N        B\ 
of  any  point  P  in  the  plane  of  the 
triangle  is  then  determined  by  finding  the  lengths  PL, 
PM,  PN  of  three  perpendiculars  dropped  from  it  upon 
the  three  fixed  lines. 

The  triangle  whose  sides  are  thus  employed  as  limits, 
is  called  the  triangle  of  reference.  The  three  perpendic- 
ulars let  fall  upon  its  sides  from  any  point,  are  termed 
the  trilinear  co-ordinates  of  the  point,  and  are  designated 
by  the  Greek  letters  «,  /?,  f. 

2OT.  On  a  first  glance,  this  system  of  co-ordinates 
seems  redundant ;  for,  in  the  Cartesian  system,  we  have 
seen  that  two  co-ordinates  are  sufficient  to  determine  a 
point;  and  it  is  obvious  from  the  diagram  that  P  is  de- 
terminable  by  any  two  of  the  perpendiculars  PL,  PM, 
PN. 

The  reader  will  therefore  not  be  surprised  to  learn  that 
the  new  method  came  into  use  as  an  unexpected  con- 
sequence of  abridging  Cartesian  equations.  The  process 
An.  Ge.  20. 


200  ANALYTIC   GEOMETRY. 

by  which  the  trilinear  system  thus  grows  out  of  the 
bilinear,  will  now  be  explained. 

2OS.  In  Art.  108,  we  have  already  hinted  at  the 
Abridged  Notation,  which  gives  rise  to  the  system  of 
which  we  are  speaking.  We  will  now  present  the  subject 
in  detail. 

If  the  equation  to  any  right  line  is  written  in  terms  of 
the  direction-cosines  of  the  line,  namely,  in  the  form 

x  cos  a  -\-  y  sin  «  — p  =  0, 

we  may  use  a  as  a  convenient  abbreviation  for  the  whole 
member  equated  to  zero ;  for  it  naturally  recalls  the  ex- 
pression into  which  it  enters  as  so  prominent  a  constant. 
Similarly,  in  the  equations 

x  cos  ft  -j-  y  sin  ft  — pr  =0, 
x  cos  f  +  y  sin  Y  — p"  =  0, 

we  may  represent  the  first  members  by  ft  and  f.  Thus 
the  equations  to  any  three  right  lines  may  be  written 

«  =  0,    0  =  0,    7*  =  0. 

The  brevity  of  these  expressions  is  advantageous,  even 
when  they  are  taken  separately ;  but  it  is  not  until  we 
combine  them,  that  the  chief  value  of  the  abridgment 
appears.  We  then  find  that  it  enables  us  to  express,  by 
simple  and  manageable  symbols,  any  line  of  a  given 
figure  in  terms  of  three  others. 

2O9.  It  is  this  last  named  fact,  which  constitutes  the 
fundamental  principle  of  trilinear  co-ordinates.  That  we 
can  so  express  a  line,  follows  from  our  being  able  (see 
Art.  108)  to  write,  in  terms  of  the  equations  to  two  given 
right  lines,  the  equation  to  any  line  passing  through  their 
intersection. 


ABRIDGED  NOTATION. 


201 


21O.  We  may  convince  ourselves  of  this,  by  a  few 

simple  examples.  If  a  =  0,  ft  =  0  represent  any  two 
right  lines,  then  (Art.  108) 

a  +  k@  =  0 

is  the  equation  to  any  right  line  passing  through  their 
intersection,  provided  Jc  is  indeterminate.  Now  &,  in 
this  equation,  (Art.  108,  Cor.  2)  is  the  negative  of  the 
ratio  of  the  perpendiculars  dropped  from  any  point  in 
the  line  a  -f-  kp  =  0  upon  the  two  lines  a  =  0  and  ft  =  0. 
Therefore,  writing  &  so  as  to  display  its  intrinsic  sign, 

a  -f  &j0  =  0 

denotes  a  right  line  passing  through  the  intersection  of 
a  =  0  and  /9  =  0,  and  lying  in  that  angle  of  the  two 
lines  which  is  external  to  the  origin  ;  but 


denotes  one  lying  in  the  same  angle  as  the  origin. 
Moreover,  when  the  perpendiculars  mentioned  are  equal, 
the  value  of  k  =  ±  1  ;  and  we  have  (Art.  109,  Cor.  3) 


the  equation  to  the  bisector  of  the  external  angle  between 
two  given  lines,  and 

a  —  fi  =  0 

the  equation  to  the  bisector  of  the  internal  angle. 

Suppose,  then,  that  we  have  a  given  triangle,  whose 
sides  are  the  three  lines 


Granting,  as  we  may,  that  the  origin  is  within  the  triangle, 
the  equations  to  the  three  bisectors  of  the  angles  will  be 


202  ANALYTIC   GEOMETRY. 

The  equations  to  the  three  lines  which  bisect  the  three 
external  angles  of  the  triangle,  will  be 


Thus,  these  six  lines  of  the  triangle  are  all  expressed  in 
terms  of  its  three  sides. 


.  We  can  of  course  extend  this  system  of  abbre- 
viations to  the  case  of  lines  whose  equations  are  in  the 
general  form 

y+C=  0, 


by  representing  the  member  equated  to  zero  by  a  single 
English  letter,  such  as  L  or  v.     Thus, 

L  -f  kl/  =  0  or  v  +  Tcv'  =  0 

denotes  a  line  passing  through  the  intersection  of  the 
lines 


It  must  be  borne  in  mind,  however,  that  in  these  equa- 
tions k  does  not  denote  the  negative  of  the  ratio  of  the 
perpendiculars  mentioned  above,  and  in  consequence 
does  not  become  ±  1  when  those  perpendiculars  become 
equal.  Hence,  in  these  cases,  the  equations  to  the  ex- 
ternal and  internal  bisectors  of  the  angle  between  two 
given  lines,  are  respectively  (see  Art.  109,  Cor.  4) 

L  +  rL'  =  0  or  lv+mv'  =  Q, 
L  —  rL'=Q  or  lv  —  mvf  =  Q, 

in  which  r,  or  m  :  I,  is  to  be  determined  by  the  formula  on 
p.  116. 

In  this  notation,  if  the  three  sides  of  a  triangle  are 

u=Q,    v  —  0,    w=Q, 


ABRIDGED  NOTATION.  203 

the  three  bisectors  of  its  interior  angles  may  be  repre- 
sented by 

lu  —  mv  =  0,     mv  —  nw  =  0,     nw  —  lu  =  0  ; 
and  the  three  bisectors  of  its  exterior  angles,  by 

lu  +  mv  =  0,     mv  -f-  nw  =  0,     ra#  -f-  lu  =  0. 

Here,  too,  the  six  lines  are  all  expressed  in  terms  of  the 
three  sides. 

212.  Having  thus  learned  how  to  interpret  the  equa- 
tions 


when  a  =  0,  /?  =  0,  f=Q  are  given  as  forming  a  tri- 
angle, we  next  advance  to  the  interpretation  of 

la  -j-  TTi/9  -{-  nf=Q  : 

an  equation  of  which  the  preceding  six  are  evidently 
particular  cases,  and  in  which  Z,  m,  n  are  any  three 
constants  whatever. 

On  the  surface,  this  looks  like  the  condition  (Art.  114) 
that  the  three  lines  a  —  0,  ft  =  0,  y=Q  shall  meet  in 
one  point.  But  the  terms  of  that  condition  are,  that 
three  lines  will  meet  in  one  point  whenever  three  con- 
stants can  be  found  such  that,  if  the  equations  to  the  lines 
be  each  multiplied  by  one  of  them,  the  sum  of  the  products 
will  be  zero.  Hence,  the  Z,  m,  and  n  of  that  condition  are 
not  absolutely  arbitrary,  but  only  arbitrary  within  the 
limits  consistent  with  causing  the  function  la  -j-  mj3  -j-  nf 
to  vanish  identically.  On  the  contrary,  the  ?,  m,  and  n 
of  the  present  equation  are  absolutely  arbitrary. 

With  this  fact  premised,  let  us  now  investigate  the 
meaning  of  the  equation. 


204  ANALYTIC  GEOMETRY. 


If  we  replace  a,  /?,  and  ?  by  the  functions  for 
which  they  stand,  and  reduce  the  equation  on  the  sup- 
position that  x  and  y  have  but  one  signification  in  all 
its  three  branches,  we  obtain 

(I  cos  a  -j-  m  cos  ft  ~f  n  cos  f)  x\ 
+  (7  sin  a  -f-  m  sin  /9  -(-  w  sin  y)  ?/  V  =  0  : 
-f  Zp  -f  ???£/  -f-  ?ip" 

which  is  evidently  the  equation  to  some  right  line,  since 
it  conforms  to  the  type 

Ax  +  By  +  C  =  Q. 

Its  full  significance,  however,  will  not  appear  until  we 
discuss  it  under  each  of  three  hypotheses  concerning  the 
relative  position  of  the  three  lines  whose  equations  enter 
it.  To  this  discussion,  we  devote  the  next  three  articles. 

214.  Let  the  three  lines  «  —  0,  /?  =  0,  f  =  0  meet  in 
one  point. 

When  this  is  the  case,  the  equation 

la  +  mft  -f-  wr  =  0 

denotes  a  right  line  passing  through  the  point  of  triple 
intersection.  For  the  co-ordinates  of  this  point  will 
render  «,  ft,  and  f  equal  to  zero  simultaneously,  and 
will  therefore  satisfy  the  equation  just  written. 

215.  Let  the  three  given  lines  be  parallel. 

In  this  case,  their  equations  (Art.  98,  Cor.)  may  be 

written 

a  =  0,     a  +  cr  =  0,     a  +  c"  =  0, 

and  the  equation  we  are  discussing  will  thus  become 

(I  -j-  m  +  n)  a  +  (mcf  +  nc")  =  0  ; 
that  is,  it  will  assume  the  form 

a  -f  c  =  0, 


a,  ft,  r,  AS  CO-ORDINATES. 


205 


and  will  therefore  denote  a  right  line  parallel  to  the  three 
given  ones. 

216.  Let  the  three  given  lines  form  a  triangle. 

In  this  case  the  preceding  results  are  avoided  ;  hence, 
the  equation  of  Art.  213  assumes  the  form 

Ax  -{-By-}-  0=0 

without  imposing  any  restriction  upon  the  values  of  A, 
B,  and  C.  In  this  case,  therefore,  it  denotes  any  right 
line  whatever. 

This  result  is  simply  the  extension  of  that  obtained  in 
Arts.  210,  211  ;  and  shows  that  we  can  (as  was  stated 
in  Art.  208)  express  any  line  of  a  figure  in  terms  of  three 
given  ones. 

217.  From  the  results  of  the  last  three  articles,  we 
conclude  that 

la  -f  mft  +  71?  =  0 

is  the  equation  to  any  right  line  ;  provided,  however, 
that  the  lines 

a  =  0,     0  =  0,     r  =  0 

are  so  situated  as  to  form  a  triangle.  This  proviso,  we 
can  not  too  carefully  remember. 

218.  The  examination  of  an  example  somewhat  more 
complex  than  any  yet  presented,  will  convince  us  that 
this  abridged  method  of  writing  equations  is  in  effect  a 
new  system  of  co-ordinates,  applicable  to  lines  of  any  order. 
Before  commencing  this  example,  it  may  be  necessary 
to  remind  the  student  that  the  use  of  a  Greek  letter  for 
an  abbreviation,  always  implies  that  the  equation  to  the 
line  so  represented  is  in  the  form 


xcosa  -|-#sina  —  p  =  0, 
and  the  like  ;  and  that,  when  the  fundamental  equations 


206  ANALYTIC  GEOMETEY. 

are  in  any  other  form,  they  will  be  abridged  by  means  of 
English  characters. 

For  the  sake  of  still  greater  brevity,  the  line  a  =  0  is 
often  cited  as  the  line  a,  the  line  f)  =  0  as  the  line  /9,  etc. 
The  point  of  intersection  of  two  lines  is  frequently  spoken 
of  as  the  point  a/9,  etc.  The  last  notation  should  be  care- 
fully distinguished  from  that  of  the  co-ordinates  of  the  same 
point. 

219.  Example. — Any  line  of  a  quadrilateral  in  terms 
of  any  three. 

Let  ABEF  be  any  quadrilateral,  and 
let 

a  =  0,     0  =  0,     7=0 

be  the  equations  to  its  three  sides  J3C, 
CA,  AB,     We  can  now  represent  any 
other  line  in  the  figure,  in  terms  of  a,  /?,       A     f 
and  7. 

Suppose  the  origin  of  bilinear  co-ordinates  to  be  within  the  tri- 
angle ABC,  and,  in  fact,  within  the  triangle  EOB.  The  equation 
to  AE,  which  passes  through  the  intersection  of  p  and  7,  and  lies 
in  their  internal  angle,  will  be  of  the  form 

mp  —  ny=0  (AE). 

The  equation  to  BF,  which  passes  through  the  intersection  of  7 
and  a,  and  lies  in  their  internal  angle,  will  be  of  the  form 

ny  —  la  =  Q  (BF). 

The  equation  to  EF,  which  joins  the  intersections  of  (a,  AE)  and 
(/?,  BF),  and  lies  in  the  external  angle  of  the  first  two  lines,  but  in  the 
internal  angle  of  the  second  pair,  must  be  formed  (Art.  108,  Cor.  2) 
so  as  to  equal  either  the  sum  of  la  and  m/3  —  ny  or  the  difference 
of  m/3  and  ny  —  In.  It  is  therefore 

la  +  mp  —  ny  =  Q  (EF). 

The  equation  to  CD,  which  joins  a/3  to  the  intersection  of  (7,  EF), 
and  lies  in  the  external  angle  of  both  pairs  of  lines,  must  be  the 
sum  of  either  la  and  m/3,  or  ny  and  la  -f-  m(3  —  ny.  Consequently, 

it  is 

la  +  mp  =  Q  (CD), 


«,  /9,  }',  AS  CO-ORDINATES.  207 

The  equation  to  OC,  which  joins  aft  to  the  intersection  of  (AE>  BF], 
and  lies  in  the  internal  angle  of  a/3,  but  in  the  external  angle  of  the 
other  two  lines,  must  be  equal  either  to  the  difference  of  la  and  m{3 
or  the  sum  of  mp  —  ny  and  ny  —  la.  Accordingly,  it  is 

la-mp=Q  (OC). 

Finally,  the  equation  to  OZ),  which  joins  the  intersection  of  (y,  EF) 
to  that  of  (AE}  BF),  and  lies  in  the  internal  angle  of  both  pairs 
of  lines,  must  be  formed  so  as  to  equal  the  difference  of  either 
la  -f  m{3  —  ny  and  ny,  or  mp  —  ny  and  ny  —  la.  Therefore,  it  is 


We  have  thus  expressed  all  the  lines  of  the  quadrilateral  in 
terms  of  the  three  lines  a,  /3,  and  y.  We  can  do  more  :  we  can 
solve  problems  involving  the  properties  of  the  figure,  by  means 
of  these  equations,  and  test  the  relative  positions  of  its  lines 
without  any  direct  reference  to  the  x  and  y  which  the  symbols  a, 
/3,  y  conceal.  Thus,  the  form  of  the  equations 

la  —  mp  =  0,     m(3  —  ny  =  0,     ny  —  la  =  0 

shows  (Art.  114)  that  the  three  lines  OC,  AE,  BF  meet  in  one 
point,  and  the  same  relation  between  OZ>,  AE,  BF  is  shown  in 
the  form  of  their  equations. 

22O.  We  see,  then,  that  by  introducing  this  abridged 
notation  we  can  replace  the  Cartesian  equations  in  x  and 
y  by  a  set  of  equations  in  «,  /9,  y.  Moreover,  it  is  notice- 
able that  all  these  abridged  equations  to  right  lines  are 
of  the  first  degree  with  respect  to  «,  /9,  and  y.  Since, 
therefore,  we  operate  upon  these  symbols  (as  the  last 
example  shows)  just  as  if  they  were  variables,  and  since 
they  combine  in  equations  which  satisfy  the  condition 
that  an  equation  to  a  right  line  must  be  of  the  first 
degree,  it  appears  that  we  may  use  a,  /?,  f  as  co-ordinates. 
Thus,  we  may  say  that  the  equation 

la  +  mp+nr  =  Q 

is  the  equation  to  any  right  line,  and  that  «,  /9,  f  are  the 
co-ordinates  of  any  point  in  the  line. 
An.  Ge.  21. 


208  ANALYTIC  GEOMETRY. 

What  now  is  the  system  of  reference  to  which  these 
co-ordinates  belong?  We  have  seen  that,  in  order  to  the 
interpretation  we  have  given  of  the  last-named  equation, 
the  line§  «,  ft,  7*  must  form  a  triangle.  We  know,  too, 
(Art.  105)  that  a  is  the  length  of  a  perpendicular  let 
fall  from  any  point  to  the  line  a  ;  that  ft  is  the  length 
of  a  perpendicular  from  the  same  point  to  the  line  ft; 
and  that  7-  is  the  length  of  a  perpendicular  from  the 
same  point  to  the  line  y.  Hence,  we  see  that  if  «,  ft,  7- 
are  taken  as  co-ordinates,  they  are  referred  to  the  tri- 
angle formed  by 


and  that  they  signify  the  three  perpendiculars  dropped 
from  any  point  in  its  plane  upon  its  three  sides.  In 
short,  we  have  come  out  upon  the  system  of  trilinear 
co-ordinates  described  in  Art.  206. 

PECULIAR   NATURE  OF  TRILINEAR  CO-ORDINATES. 

221.  Before  making  any  further  application  of  the 
new  system,  it  is  important  to  notice  that  trilinear  co- 
ordinates are  in  one  respect  essentially  different  from 
bilinear.  In  the  Cartesian  system,  the  x  and  y  are 
independent  of  each  other,  unless  connected  by  the 
equation  to  some  locus.  In  the  trilinear  system,  on  the 
contrary,  the  «,  ft,  and  7  are  each  of  them  determined 
by  the  other  two  ;  that  is,  there  is  a  certain  equation 
between  them,  which  holds  true  in  all  cases,  whether  the 
point  which  they  represent  be  restricted  to  a  locus  or  not. 
In  the  language  of  analysis,  we  express  this  peculiarity 
by  saying  that  each  of  these  co-ordinates  is  a  determined 
function  of  the  other  two. 

That  trilinear  co-ordinates  are  subject  to  such  a  con- 
dition, is  proved  by  the  fact,  that  two  co-ordinates  com- 


TRILINEARS  ALWAYS  CONDITIONED.          209 

pletely  fix  a  point,  and  must  therefore  determine  the 
value  of  any  third.  The  equation  expressing  the  exact 
limits  of  the  condition,  we  now  proceed  to  develop. 

222.  Let  ABC  be  the  triangle  of  reference,  a  =  0 
being  the  equation  to  the  side  BC, 
y3  =  0  the  equation  to  CA,  and  f  =  0 
the  equation  to  AB.  Then,  if  P  be 
any  point  in  the  plane  of  the  triangle, 
the  three  perpendiculars  PL,  PM,  N, 

PN  will  be  represented  by  «,  /9,  f. 
Supposing  now  that  a,  6,  and  c  denote  the  lengths  of  the 
three  sides  BC,  CA,  AB,  we  shall  have 

aa  =  twice  the  area  of  BPC, 
bp  =  «  .  «  CPA, 
cf  =  "  "  APE. 

But  the  sum  of  these  areas  is  constant,  being  equal  to 
the  area  of  the  triangle  of  reference ;  therefore,  repre- 
senting the  double  area  of  this  triangle  by  M,  we  obtain 

aa  -j-  6/9  -f  cy  —  M: 

which  is  the  constant  relation  connecting  the  trilinear 
co-ordinates  of  any  point. 

Remark — When  we  say  that  the  sum  of  the  areas  of  APE, 
BPC,  CPA  is  equal  to  the  area  of  ABC,  we  of  course  mean  their 
algebraic  sum.  For,  if  we  take  a  point  outside  of  the  triangle  of 
reference,  such  as  P',  we  shall  evidently  have 

ABC=  APfB  -f-  BP'C—  CP'A  • 

that  is,  in  such  a  case,  one  of  the  three  areas  becomes  negative. 
And  this  is  as  it  should  be ;  for,  as  we  have  agreed  to  suppose  the 
Cartesian  origin  to  be  WITHIN  the  triangle  of  reference,  the  perpen- 
dicular P/M  is  negative,  according  to  the  third  corollary  of  Art.  105. 


210  ANALYTIC   GEOMETRY. 

223.  This  peculiarity  of  trilinear   co-ordinates  may 
be  made  to  promote  the  advantages  of  the  system. 

In  the  first  place,  it  gives  rise  to  a  very  symmetrical 
expression  for  an  arbitrary  constant.  For  if  k  be  arbi- 
trary, kM  will  also  be;  and  we  shall  have,  for  the 
symbol  alluded  to, 

k(aa  +  bp  +  cr)  (1). 

We  can  also  modify  this  symbol,  and  render  it  still 
more  useful.  Let  1  :  r  =  the  ratio  of  the  side  a,  in 
the  triangle  of  reference,  to  the  sine  of  the  opposite 
angle  A.  Then,  by  Trig.,  867,  we  shall  have 

sin  A       sin  B      sin  0 
a  b  c 

Now,  by  the  principle  of  Art.  222, 

r  (aa  -f  5/3  -j-  cf)  =  constant  ;  , 

hence,  substituting  for  ra,  rb,  re  from  the  equal  ratios 

above, 

a  sin  A  -f-  /3  sin  B  -f-  f  sin  C  =  constant  ; 

or,  the  symbol  for  an  arbitrary  constant  may  be  written 
k  (a  sin  A  +  /3  sin  B  -f-  r  sin  0)  (2). 

224.  In  the  second  place,  the  peculiarity  of  trilinears 
enables    us   to   use  homogeneous  equations  in   all  cases. 
For,  if  a  given  trilinear  equation  is  not  homogeneous, 
we  can  at  once  render  it  so  by  means  of  the  relation  in 
Art.  222.     Thus,  if  the  given  equation  were  p=p,  we 
might  write 


225.  In  the  third  place,  instead  of  the  actual  trilinear 
co-ordinates  of  a  point,  we  may  employ  any  three  quan- 
tities that  are  in  the  same  ratio,  without  affecting  the 


TRANSFORMATION  TO  BILISEARS.  211 

equation  to  the  locus  of  the  point.  For,  since  all  tri- 
linear  equations  are  homogeneous,  the  effect  of  replacing 
«,  /3,  Y  by  />«,  /9/9,  pf  will  only  be  to  multiply  the  given 
equation  by  the  constant  p  :  which  of  course  leaves  it 
essentially  unchanged. 

This  principle  will  often  prove  of  great  convenience. 

Remark.  —  In  case  it  becomes  desirable  to  find  the  actual  tri- 
linear  co-ordinates  when  they  have  been  displaced  in  the  manner 
described,  we  may  proceed  as  follows  :  —  Let  I,  m,  n  be  the  three 
quantities  in  a  constant  ratio  to  a,  /?,  y:  then,  supposing  the  ratio 
to  be  1  :  r, 

a  =  rl,     P  =  rm,     y  =  rn. 
Hence,  (Art.  222,) 

r  (la  -f-  mb  +  nc}  =  M: 

from  which  r  is  readily  found. 

226.  Finally,  the  property  in  question  enables  us  to 
pass  from  trilinear  co-ordinates  to  Cartesian.  For,  by 
means  of  the  relation  aa-\-bft-{-cf  —  M,  we  can  convert 
any  trilinear  equation  into  one  which  shall  contain  only 
ft  and  f.  Then,  supposing  the  side  f  of  the  triangle  of 
reference  to  be  the  axis  of  x,  and  the  side  /9  the  axis  of  ?/, 
we  shall  have 

ft  =  x  sin  A,     f  =  y  sin  A. 

Corollary.  —  If  the  triangle  of  reference  should  be  right- 
angled  at  A,  the  reduction-formulae  will  become 


But,  in  general,  to  pass  from  trilinears  to  rectangulars 
when  the  side  f  is  taken  for  the  axis  of  x,  we  must  use 

ft  =  xs'mA  —  y  cos  A,       f  =  y. 

The-  student  may  draw  a  diagram,  and  verify  the  last 
formulae. 


212  ANALYTIC  GEOMETRY. 


The  advantage  of  the  trilinear  system  consists 
in  this  :  Its  equations  may  all  be  referred  to  three  of 
the  most  prominent  lines  in  the  figure  to  which  they 
belong,  and  hence  become  shorter  and  more  expressive 
than  those  of  the  Cartesian  system,  which  can  go  no 
farther  in  the  process  of  simplification  than  the  use  of 
two  prominent  lines.  The  student  will  see  a  good  illus- 
tration of  this  in  the  equations  to  the  bisectors  of  the 
internal  angles  of  a  triangle.  The  trilinear  equations 
are 

«  —  0  =  0,     /9  —  r  =  0,     r~  «  =  0 

which  are  much  simpler  than  the  Cartesian  ones  given 
in  Ex.  32,  p.  124. 

TRILINEAR  EQUATIONS  IN  DETAIL. 

228.  Equation  to  any  Right  Uiie.—  This  we  have 
already  (Arts.  212—217)  found  to  be 

la  +  mfi  +  nr  =  Q. 

229.  Equation  to  a  right  line  parallel  to  a  given 

one.  —  It  is  obvious  geometrically,  that  the  «,  /?,  7-  of  the 
parallel  line  will  each  differ  from  the  a,  /9,  7  of  the  given 
one  by  some  constant.  Hence,  the  given  line  being 
la  -j-  mf!  4-  nf  =  0,  the  required  equation  [Art.  223,  (2)] 
will  be 

la  +  mfi  +  ny  +  k  (a  sin  A  -f  /?  sin  B  -f  7  sin  C)  =  0. 

230.  Equation  to  a  right  line  situated  at  in- 
finity. —  The    Cartesian   equation  to   this   is   (7=0,   in 
which    (Art.    110)    C  is    a    finite    constant.     Therefore 
(Art.  223)  the  trilinear  equation  is 

a  sin  A  -f-  /9  sin  B  +  7  sin  C=  0. 


TEILINEAR  EQUATIONS. 


213 


L.  Condition  that  two  right  lines  shall  be 
mutually  perpendicular. — Let  the  two  lines  be 

la  +  mp  +  nr  =  0,     Va  +  m'ft  -j-  n'r  =  0. 

By  writing  «,  /?,  7-  in  full,  collecting  the  terms,  and  apply- 
ing (Art.  96,  Cor.  3)  the  criterion  AAr  -f  BB1  =  0,  we 
obtain,  as  the  required  condition, 

(mnf  -f-  m'n)  cos  (/9  —  7-)  \ 

(nlr    -f-   n'Z)  cos  (7*  —  a)  >•  =  0. 

(?m;  -f-  Z'm)  cos  («  —  /5)  J 

Now,  in  this  expression,  «,  /?,  7*  are  the  angles  made  with 
the  axis  of  x  by  perpendiculars 
from  the  origin  on  the  lines  «,  /9,  f. 
Supposing  the  latter  to  form  the 
triangle  of  reference,  and  to  inclose 
the  origin,  it  is  evident  from  the 
diagram,  that  /?  —  7-  is  the  angle 
between  the  perpendiculars  /?  and 
f ;  that  f  —  a  is  the  angle  between 
the  perpendiculars  f  and  a;  and  that  a. 
between  the  perpendiculars  a  and  /?.  From  the  proper- 
ties of  a  quadrilateral  it  then  follows,  that  /9  —  ^  is  the 
supplement  of  A9  f  —  a  of  B,  and  a  —  /9  of  C.  Hence, 
the  required  condition  may  be  otherwise  written 


X'" 


is  the  angle 


mm 


((mn'  +  m'r. 

nn'  —  •<  (nlf    -f    n' 

[  (lmr  -f  I'rt 


n)  cos  A 
7)  cos  B 

I'm)  cos  (7 


Corollary.  —  The  condition  that  la  -f-  mft  -f 
be  perpendicular  to  f  =  0,  is 


=  0. 


=  0  shall 


n  =  m  cos 


I  cos 


214  ANALYTIC  GEOMETRY. 

2320  Length  of  a  perpendicular  from  any 
point  to  a  given  right  line.  —  Let  the  point  be  0/3^, 
and  the  line  la  -j-  mft  -f  ^T  =  0-  Write  the  latter  equa- 
tion in  full,  collect  its  terms,  and  apply  the  formula  of 
Art.  105,  Cor.  2.  We  thus  find 


p_    _ 

l/(72-|-m2-f-^2  —  2mn  cosA  —  2nl  cos.5  —  2lm  cos  (7) 

233.  [Equation  to  a  right  line  passing  through 
Two  Fixed  Points.  —  The  form  of  this  will  of  course 
be 

la  +  mp  -f  wr  =  0  ; 

and  our  problem  is,  to  determine  the  ratios  l:m:  n  so 
that  the  line  shall  pass  through  the  two  points  «1/?1?'1, 


Since  the  two  points  are  to  be  on  the  line,  we  shall 
have 

Z«!  -f  mfc  -f  wf,  =  0, 

Ia2  +  mft2  +  nf2  =  0. 

Solving  for  ?  :  n  and  m  :  n  between  these  conditions,  we 
find 


:  m  :  n  = 
The  required  equation  is  therefore 

«  far*  —  r&)  +  ^  (n«2  -  «ir2)  -f  r  («  A  —  A«2)  =  o. 

Corollary,  —  Hence,   in    trilinears,    the    condition    that 
three  points  shall  lie  on  one  right  line  is 

—  n&)  -f  A  (Ti«2  —  «ir2  -f  3  «i    —   «  =  o. 


Expanding,  and  re-collecting  the  terms,  we  may  write 
this  more  symmetrically  (see  Art.  112), 

«2  (An  —  rA)  4- 


TRIUNE AR  EQUATIONS. 


215 


234.  Theorem. — Every  trilinear  equation  of  the  second 
degree  represents  a  conic. 

For,  since  «,  ft,  f  are  linear  functions  of  x  and  y, 
every  such  equation  is  reducible  to  an  equation  of  the 
second  degree  in  x  and  y.  But  the  latter  (Art.  203) 
will  represent  a  conic. 

Remark. — In  passing  now  to  the  trilinear  expressions 
for  curves  of  the  Second  order,  we  shall  at  first  suppose 
the  triangle  of  reference  to  have  a  special  position,  such 
as  will  tend  to  simplify  the  resulting  equations.  The  more 
general  equations,  corresponding  to  any  position  of  the 
triangle,  will  be  investigated  afterward. 

235.  Equation  to  the  Conic,  referred  to  the 
Inscribed  Triangle. — To  obtain  this,  we  must  form 
an  equation  of  the  second  degree  in  «,  ft,  f,  such  as  the 
co-ordinates  of  the  vertices  of  the  reference-triangle  will 
satisfy. 

Now,  at  the  vertex  A,  we  have 
ft  =  0,  Y  =  0  ;  at  the  vertex  B,  Y  =  0, 
a  =  0  ;  and,  at  the  vertex  C,  a  —  0, 
ft  =  0.  Hence,  the  required  equation 
may  be  written 

-f  naft  =  0  ; 


for  this  expression  is  obviously  satisfied  by  either  of  the 
three  suppositions 

Corollary. — Dividing   through   by  aftf,  we   may  write 
the  equation  just  found  in  the  more  symmetrical  form 


m 


n 


216  ANALYTIC  GEOMETRY. 

23G.  Equation  to  the  Circle,  referred  to  the 
Inscribed  Triangle.  —  Our  problem  here  is,  to  deter- 

mine I  :  m  :  n  so  that 


shall  represent  a  circle. 

Write  a,  /9,  7-  each  in  full,  expand  the  equation,  collect 
the  terms  with  reference  to  x  and  ?/,  and  apply  the  cri- 
terion (Art.  140)  H=Q,  A  —  B  =  0.  The  conditions 
in  order  that  (1)  may  represent  a  circle  will  thus  be 
found  to  be 

I  cos  (j9  -\-  Y)  -\-  m  cos  (f  +  «)  -f-  7i  cos  (a  -f-  /?)  =  0, 

Z  sin  (/3  +  f)  +  w  sin  (7-  -f-  a)  -f-  w  sin  (a  +  /9)  —  0. 

Solving  these  for  Z  :  n  and  m  :  w,  and  applying  Trig.,  845, 
in,  we  obtain 

I  :  m  :  n  =  sin  (ft  —  f)  '•  sin  (/  —  °)  :  s^n  (a  —  $)• 


But  (see  Art.  231)  ft  —  T  =  ISO°—  A,  r  —  a=ISQ°—B, 
a  —  ft  =  180°  —  C.     Hence,  the  equation  sought  is 

sin  A       sin  B       sin  C n 

\  -Q        ~T~ =  ". 

Corollary. — The  equation  to  any  circle  concentric  with 
the  one  circumscribed  about  the  triangle  of  reference,  will 
only  differ  from  the  foregoing  (Art.  142,  Cor.  1)  by  some 
constant.  Hence,  it  may  be  written 

sin  A    .sin  B       sin  C       ,  ,      .  n   •     T> 

1 r—  -\ —  A;  (asm  A -{- ft  sm  B -{- ?  sm  C). 

237.  General    equation    to    the    Circle.  —  The 

equation  of  the  preceding   article   applies   only  to  the 


TRILINEAR  EQUATIONS.  217 

circle  described  about  the  triangle  of  reference.  We 
are  now  to  seek  an  equation  which  will  represent  any 
circle. 

The  rectangular  equation  to  a  circle  (Art.  140)  may 
be  put  into  the  form 

X*        _f-          ^          +         AX         _|_       ty        ^          C     =         Q 

In  this,  the  only  arbitrary  constants  are  .A,  B,  C. 
Hence,  rectangular  equations  to  different  circles  will 
vary  only  in  the  linear  part,  and  the  equation  to  any 
circle  may  be  formed  from  that  of  a  given  one  by  merely 
adding  to  the  latter  an  arbitrary  linear  function.  Now 
this  property  is  as  true  of  trilinear  as  of  rectangular 
equations,  since  a  trilinear  equation  is  only  a  rectangu- 
lar one  written  in  a  peculiar  way.  We  can  therefore 
form  the  equation  to  any  circle  from  that  of  the  circle 
described  about  the  triangle  of  reference,  by  adding  to 
the  latter,  terms  in  the  type  of  la  -f-  vnft  -j-  nf. 

By  clearing  of  fractions,  and  replacing  sin  A,  sin  B, 
sin  C  by  «,  6,  c,  which  are  in  the  same  ratio,  the  equation 
to  the  circumscribed  circle  becomes 

a?r  +  b?a  -f  oafi  =  0. 
Hence,  the  required  equation  is 


in  which  M  is  the  fixed  constant  a  a  -}-  5/9  -f-  <?^,  and  is 
multiplied  into  the  linear  function  in  order  to  render  the 
equation  homogeneous. 

Remark,  —  When  convenience  requires  it,  we  can  replace  M 
(Art.  223)  by  the  constant  a  sin  A  -f-  /5  sin  B  -f  7  sin  (7,  and  write 
the  equation 


218  ANALYTIC  GEOMETRY. 

238.  Triliiiear  equation  to  tlte  Conic  in  Gen- 
eral. —  From  Arts.  224,  234,  it  follows  that  this  is  simply 
the  general  homogeneous  equation  of  the  second  degree 
in  «,  /?,  /-.  We  therefore  write  it 


Cf  -j-  2Ffr  +  26>  +  2jffa0  ==  0. 

Kemark.  —  It  is  worthy  of  notice,  that,  if  we  suppose  7=1,  this 
expression  becomes 


Aa?  -f  2Ha{3  +  B(P  +  2Ga  +  2F/3  +  C=  0, 
and  is,  in  form,  identical  with 

Ax1  +  2Hxy  +  By*  +  2Gx  +  2Fy+C=  0, 


the  Cartesian  equation  to  the  Conic  in  General.  This  fact  has  led 
Salmon  to  the  opinion  that  Cartesian  co-ordinates  are  a  case  of 
triUnear.* 

239.  Problem,  —  To  determine  the  condition  in  order 
that  the  general  triUnear  equation  of  the  second  degree 
may  represent  a  circle. 

From  the  equation  of  Art.  222,  it  is  evident  that  we  have  the 
following  relations  : 

aa2  =  Ma  —  6  a/3  —  cya, 
bp  =  Mi3  —  c/ty  —  ao/3, 
cf  —  My  —  aya  —  bfiy. 

Substituting  from  these  for  a2,  /32,  y2  in  the  equation  of  Art.  238,  we 
find  that  it  may  be  written 


*  See  his  Conic  Sections,  p.  67,  4th  edition ;  but  compare  the  principles 
of  Arts.  221,  222. 


TEILINEAE  EQUATIONS.  219 

Comparing  this  with  the  equation  to  the  circle, 

a(3y  +  by  a  +  cap  +  M  (la  +  m{3  +  ny  )  =  0, 
we  see  that  it  will  represent  a  circle,  provided 


a  b 

Hence,  the  required  condition  is 

—  Ca2  -  Ac1  = 


24O.  Equations  to  the  Chord  and  the  Tangent 
of  any  Conic. — For  the  sake  of  simplicity,  we  shall 
suppose  the  inscribed  triangle  to  be  the  triangle  of 
reference. 

I.  The  equation  to  the  chord  is  the  equation  to  the 
right  line  which  joins  any  two  points  on  the  curve,  as 
a'ft'f,  a."finf.  Now  since  these  points  are  on  the  conic, 
we  shall  have  (Art.  235,  Cor.) 

I       Tti      n  I         m        n 

a'      j3f      f         '          a77       /9"~  ~~  p7  ~ 

We  can  now  easily  see  that  the  equation  to  the  chord 
may  be  written 

la         mfi         nr 


ff 

For  this  is  the  equation  to  a  right  line,  since  it  is  of  the 
first  degree ;  and  to  the  line  that  passes  through  a'ft'f, 
a"fi"f,  since  it  is  satisfied  by  the  co-ordinates  of  either 
point,  inasmuch  as  they  convert  it  into  one  of  the  con- 
ditions just  found  above. 

II.  The  point  a"fttrff  may  be  supposed  to  approach 
the  point  a'^f  as  closely  as  we  please.  At  the  moment 
of  coincidence,  the  chord  becomes  the  tangent  at  a' 


220  ANALYTIC  GEOMETRY. 

and  we  also  have  a"—  a',  /3"  =  /9',  f  =  f.  By  making 
the  corresponding  substitutions  in  the  equation  to  the 
chord,  we  shall  therefore  obtain  the  equation  to  the 
tangent.  It  is 

la  J.  ™P  -il  nr       n 

^-H-^r-+pj- 

241.  There  are  many  other  equations,  more  or  less 
symmetrical,  representing  the  Conic  or  the  Circle,  but 
our  limits  forbid  their  presentation.  Those  already 
developed  have  the  widest  application,  and  are  sufficient 
to  illustrate  the  trilinear  method.  The  student  who 
wishes  further  information  on  the  subject,  may  consult 
SALMON'S  Conic  Sections. 

EXAMPLES. 

1.  When  will  the  locus  of  a  point  be  a  circle,  if  the  product 
of  its  perpendiculars  upon  two  sides  of  a  triangle  is  in  a  constant 
ratio  to  the  square  of  its  perpendicular  on  the  third? 

Take  the  triangle  mentioned,  as  the  triangle  of  reference  ;  and  repre- 
sent the  constant  ratio  spoken  of,  by  k.  The  conditions  then  give  us 


so  that  the  locus  is,  in  general,  a  conic  of  some  form.  In  order  that  it 
may  be  a  circle,  the  equation  just  found  must  satisfy  the  condition  of 
Art.  239.  Applying  this,  and  observing  the  fact  that  in  the  present 
equation  C=  —  k,  2H=  1,  and  A,B,F,G  each  =  0,  we  find  that  the  locus 
will  be  a  circle  if 

kb*  =  ka*  =  ab. 

From  the  first  of  these  conditions,  we  obtain  a=  b;  from  the  second  and 
third,  k  =  l.  The  required  condition  then  is,  that  the  triangle  shall  be 
isosceles,  and  the  constant  ratio  unity. 

It  is  interesting  to  notice  how  clearly  the  equation  above  written 
expresses  the  position  of  the  locus  with  respect  to  the  triangle.  To  find 
where  the  side  a  cuts  the  conic,  we  simply  make  (Art.  62,  Cor.  1)  a^  0 
in  the  equation 

a/?  =  ky*. 

The  result  is  y2=  0  ;  that  is,  the  equation  whose  roots  are  the  co-ordinates 
of  the  points  of  intersection  is  a  quadratic  with  equal  roots.  Hence, 


CONIC  BY  TWO  TANGENTS  AND  CHORD.       221 

(Art.  62,  3d  paragraph  of  Cor.  2,)  the  line  a  meets  the  curve  in  two  coin- 
cident points  ;  in  other  words,  it  touches  the  conic.  Moreover,  since  the 
quadratic  of  intersection  is  y2  =  0,  the  two  points  coincide  on  the  line  y. 
We  thus  learn  that  the  conic  touches  the  side  a  of  the  triangle  at  the 
point  ya. 

If  we  make  /?=  0  in  the  equation,  we  again  obtain  y2  =  0.  Hence,  the 
conic  touches  the  side  /?  of  the  triangle  at  the  point  /?y.  If  we  make 
y  =  0,  we  get  either  a  =  0  or  /3  =  0  ;  the  side  y  therefore  cuts  the  conic  in 
two  points  :  one  on  the  side  a,  the  other  on  B ;  or,  one  the  point  ya,  the 
other,  the  point  /?y. 

We  may  sum  up  the  whole  result  by  saying  that  a/?  =  fcy2  denotes  a 
conic,  to  which  two  of  the  sides  of  the  triangle  of  reference  are  tangents,  while 
the  third  is  a  chord  uniting  their  points  of  contact. 

2.  Form  the  trilinear  equations   to  the  right  lines  joining  the 
vertices  of  a  triangle  to  the  middle  points  of  the  opposite  sides. 

Take  the  triangle  itself  as  the  triangle  of  reference,  and  suppose  the 
Cartesian  origin  within  it.  The  required  equations  (Art.  108,  Cor.  2)  will 
be  of  the  form 

a-fc/?  =  0,      /3-k'-y=Q,      y-k"a=0: 

in  which  we  are  to  determine  k,  k' ,  k"  so  that  the  lines  shall  bisect  the 
sides  of  the  triangle. 

We  know  that  k  is  =  the  ratio  of  the  perpendiculars  dropped  from 
the  middle  of  y  on  a  and  /?  respectively.  But  this  ratio  is  evidently 
=  sin  B  :  sin  A.  Similarly,  k'  —  sin  C  :  sin  B ;  and  k"  =  A  :  sin  (7. 
Hence,  the  equations  sought  are 

a  sin  A  — 13  sin  B  =  0,     Bain  B  —  y  sin  C=  0,     y  sin  C—  a  sin  A  =  0. 

3.  Form  the   trilinear  equations    to    the    three    perpendiculars 
which  fall  from  the  vertices  of  a  triangle  upon  the  opposite  sides. 

We  begin,  as  before,  with  the  forms  of  the  equations,  namely, 
a-fc/?=0,  p-k'y=Q,  y-fc"a=0.  Now  the  condition  that  the  first 
line  shall  be  perpendicular  to  y  (Art.  231,  Cor.)  gives  us 

cos  B —  A;  cos  ^4  =  0. 

Hence,  k  —  cos  B  :  cos  A;  and,  similarly,  k'=  cos  C:.  cos  B,  k"  =  cos  C:  cos  A. 
The  required  equations  are  therefore 

a  cos  A  —  (3  cos  B  =  0,     /?  cos  B  —  y  cos  C=  0,     y  cos  C—  a  cos  A  =  0. 

4.  Form    the    trilinear    equations   to  the   three    perpendiculars 
through  the  middle  points  of  the  sides  of  a  triangle. 

The  middle  points  of  the  sides  may  be  regarded  as  the  intersections 
of  y,  a,  and  0  with  the  lines  of  Ex.  2.  Hence,  the  for m  of  the  equations 
will  be 

— /?sin#+ny=0,    0  sin  B—  y  sinC+la  =  0,     y  sin  C—  a  sin  A+  m/3  =  0. 


222  ANALYTIC  GEOMETRY. 

But  the  condition  of  Art.  231,  Cor.,  gives  us  n  =  sin  (A—B),  l—sm(B—  C), 
m  =  sin  (  C—  A).     Therefore,  the  equations  sought  are 

a  sin  A  —  0  sin  B  +  y  sin  (A  —  B)  =  0, 
(3  sin  B  -  y  sin  C  +  a  sin  ( B  —  C)  =  Q, 
y  sin  C-  a  sin  A  +  B  sin  (  C-  A)—-0. 

If  the  student  will  now  compare  the  equations  of  the  last  three  ex- 
amples with  those  of  Exs.  4,  6,  7  on  page  121,  he  will  at  once  see  how 
much  simpler  the  trilinear  expressions  are  than  the  Cartesian. 

5.  Show  that  the  equation  representing  a  perpendicular  to  the 
base  of  a  triangle  at  its  extremity,  is  a  +  ycos-B=  0. 

6.  Show    that    the    lines    o  — A-/?  =  0,  j3—ka  =  Q    are    equally 
inclined  to  the  bisector  of  the  angle  between  a  and  (3. 

1.  Prove  that  the  equation  to  the  line  joining  the  feet  of  two 
perpendiculars  from  the  vertices  of  a  triangle  on  the  opposite  sides  is 

a  cos  A  +  P  cos  B  —  y  cos  (7=0. 

Also,  that  the  equation  to  a  line  passing  through  the  middle  points 
of  two  sides  is 

a  sin  A  +  /3  sin  B  —  y  sin  C=Q. 

8.  Show  that  the  equation  to  a  line  through  the  vertex  of  a 
triangle,  parallel  to  the  base,  is  a  sin  A  -f  (3  sin  .8=0. 

9.  Find  the  equation  to  the  line  which  joins  the  centers  of  the 
inscribed  and  circumscribed  circles  belonging  to  any  triangle. 

By  the  principle  of  Art.  225,  we  may  take  for  the  co-ordinates  of  the 
first  point  1,  1,  1 ;  and,  of  the  second,  cos  A,  cos  B,  cos  C.  Hence, 
(Art.  233,)  the  equation  is 

a  (cos  B  —  cos  C)  +  0(cos  C—  cos  A)  +  y  (cos  A  —  cos  B)  =  0. 

10.  What  is  the  locus  of  a  point,  the  sum  of  the  squares  of  the 
perpendiculars  from  which  on  the  sides  of  a  triangle  is  constant? 
Show  that  when  the  locus  is  a  circle,  the  triangle  is  equilateral. 

11.  Find,  by  a  method  similar  to  that  of  Art.  231,  the  tangent 
of  the  angle  contained  by  the  lines 

la  -f  m/3  +  ny  =  0,    I' a  -f  m'/3  +  n'y  =•  0. 

12.  Write  the  equation  to  the  circle  circumscribing  the  triangle 
whose  sides  are  3,  4,  5. 

13.  A  conic  section  is  described  about  a  triangle  ABC;  lines 
bisecting  the  angles  A,  B,  and  C  meet  the  conic  in  the  points  A/ , 
B',  and  C/ :  form  the  equations  to  A'B,  AfC,  A'B'. 


INSCRIBED  AND  ESCRIBED  CONICS.  223 

14.  Find  an  equation  to  the  conic  which  touches  the  three  sides 
of  a  triangle. 

If   the  equations  to  the  three  sides  of  the   triangle  are  a  =  0,  /?  —  0, 
y  —  0,  the  required  equation  may  be  written 


Verify  this,  by  clearing  of  radicals,  and  showing  that  y,  a,  and  /?  are  alt 
tangents  to  the  curve.     (See  Art.  240.) 

Note,  —  The  preceding  equation  is  of  great  importance  in  some  investi- 
gations, and  it  will  be  found  upon  expansion  to  involve  four  varieties  of  sign, 
in  the  terms  containing  /?y,  ya,  a/?.  This  agrees  with  the  fact  that  there 
are  four  conies  which  touch  the  sides  of  a  triangle,  namely,  one  inscribed, 
and  three  escribed  —  that  is,  tangent  to  one  side  externally,  and  to  the 
prolongations  of  the  other  two  internally.  If  we  suppose  a,  0,  y  all  pos- 
itive, therefore,  the  equation  will  represent  the  inscribed  conic  ;  thus, 


The  equations  to  the  escribed  conies  will  be 

y^Ti  +  Vlntf  +  y^  =  0,      Via  +  y^n~/3  +  V^=  0,      VTa  +  V^f3  +  V^  =  0, 


since,  in  each,  one  set  of  perpendiculars  must  fall  on  the  triangle  exter- 
nally. 

15    Find  the  equation  to  the  circle  inscribed  in  any  triangle. 

We  may  derive  this  from  Via  +  Vm\l  +  J^iy^O,  by  clearing  of  radicals 
on  the  assumption  that  a,  /?,  y  are  positive,  and  then  taking  I,  m,  n  so  as 
to  satisfy  the  condition  of  Art.  239.  Or,  we  may  develop  the  equation 
from  that  of  the  circumscribed  circle,  as  follows*:  —  Let  the  sides  of  the 
triangle  formed  by  joining  the  points  of  contact  of  the  inscribed  circle  be 
a',  0',  y'.  Its  equation  (Art.  236)  will  then  be 

sin  A'       sin  B'       sin  C' 

~~^~    ~&~    ~7~  =0- 

But,  with  respect  to  the  triangles  AB'C',  etc.,  -we  have  (Ex.  1)  a'2=/?y, 
/?'2  =  ya,  y'2=a/?;  and  A'=  90°  -  Y2  A,  B'  =  90°  -  %  B,  C'=W°-%C. 
Substituting  these  values,  and  multiplying  the  resulting  expression 
throughout  by  ^a/j/y,  the  required  equation  is  found  to  be 

cos  M  A  ^a  +  cos  }£  B  VJ+  cos  K  CfV'7=  0. 

The  student  may  now  investigate  the  equations  to  the  three 
escribed  circles.  By  the  principle  developed  in  Ex.  14,  Note,  these 
may  be  written  down  at  once,  from  the  equation  just  found. 


*  Hart's  solution,  quoted  by  Salmon. 
An.  Ge.  22. 


224  ANALYTIC  GEOMETRY. 


SECTION  II.  —  TANGENTIAL  CO-ORDINATES. 

242.  We  now  come  to  a  method  of  representing  lines 
and  points,  which,  in  connection  with  the  trilinear,  plays 
a  very  important  part  in  the  Modern  Geometry.  It  is 
known  as  the  method  of  Tangential  Co-ordinates,  and  was 
first  employed  by  MOBIUS,  in  his  Barycentrisclie  Calcul, 
which  appeared  in  1827. 

We  can  here  give  only  an  outline  of  the  system.  For 
a  full  exhibition  of  it  in  its  most  important  applications, 
the  student  is  referred  to  the  work  just  cited,  to  Pliicker's 
System  der  analytischen  Geometric,  and  to  Salmon's  Conic. 
Sections. 

218.  The  following  considerations  will  bring  into 
view  the  relations  of  the  new  system  to  the  Cartesian 
and  the  trilinear. 

Let  AX  +  py  +  v  =  0  be  the  equation  to  a  right  line. 
Since  the  position  of  the  line  is  determined  by  fixing  the 
values  of  /,  //,  v,  it  is  evident  that  we  may  regard  these 
co-efficients  as  the  co-ordinates  of  a  right  line. 

Suppose  then  we  take  ^  /jt,  v  as  variables,  and  connect 
them  by  any  equation  of  the  first  degree,  «/-j-5/^-j-cv=0. 
Such  an  equation  (Art.  115,  Cor.)  is  the  condition  that 
the  whole  system  of  lines  denoted  by  /#  +  p.y  -f-  v  =  0 
shall  pass  through*a  fixed  point,  whose  co-ordinates  are 
a  :  c,  b  :  c.  Since,  then,  the  point  becomes  known 
whenever  ah  +  b/jt  -|-  cv  =  0  has  known  co-efficients,  we 
must  regard  the  condition  just  written  as  the  equation 
to  a  point. 

Accepting  this  result,  and  putting  «,  /?,  etc.,  as  abbre- 
viations for  the  expressions  equated  to  zero  in  the  equa- 
tions to  different  points,  we  shall  have,  by  the  analogy 
of  Art.  108,  la  -f  mfi  =  0  as  the  equation  to  a  point 


TANGENTIAL  CO-OEDINATES.  225 

dividing  in  a  given  ratio  the  distance  between  the  points 
a  =  0,  £  =  0.     Similarly, 

la  —  mft  =  0,     mft  —  ny  —  0,     ny  —  la  =  0 
denote  three  points  which  lie  on  one  right  line. 

244.  From  what  has  just  been  said,  we  can  see  that 
we  may  have  a  system  of  notation  in  which  co-ordinates 
represent  rigid  lines,  while  points  are  represented  ly  equa- 
tions of  the  first  degree  between  the  co-ordinates.  This  is 
what  we  mean  by  a  system  of  tangential  co-ordinates. 

The  tangential  method  is  in  a  certain  sense  the  recip- 
rocal of  the  Cartesian.  It  begins  with  the  Right  Line, 
and,  by  means  of  an  infinite  number  of  right  lines  all 
passing  through  the  same  point,  determines  the  Point; 
the  Cartesian  method,  on  the  contrary,  begins  with  the 
Point,  and  determines  the  Kight  Line  as  the  assemblage 
of  an  infinite  number  of  points.  In  the  tangential  sys- 
tem, accordingly,  the  Right  Line  fulfills  the  office  assigned 
to  the  Point  in  the  Cartesian  :  it  is  the  determinant  of  all 
forms,  which  are  conceived  to  be  obtained  by  causing  an 
infinite  number  of  right  lines  to  intersect  each  other  in 
points  infinitely  close  together. 

345.  Article  243  has  shown  us  that  Cartesian  co- 
efficients are  tangential  co-ordinates,  and  that  the  tan- 
gential equation  of  the  first  degree  is  a  Cartesian  equation 
of  condition.  We  shall  presently  see  that  tangential 
equations  are  always  Cartesian  conditions,  and  in  fact 
signify  that  a  right  line  passes  through  two  consecutive 
points  on  a  given  curve;  that  is,  through  two  points 
infinitely  near  to  each  other;  and  so,  is  a  tangent  to  the 
curve. 

246.  The  truth  of  this  will  appear  from  the  following 
geometric  interpretation  of  tangentials,  which  will  bring 


226  ANALYTIC  GEOMETRY. 

the  system  into  relation  to  curves  of  all  orders,  and  show 
that  we  can  represent  any  curve  in  this  notation. 

Let  AS,  CD,  EF  be  any  three 
lines,  each  passing  through  two 
consecutive  points  of  the  curve 
LM,  and  therefore  touching  it, 
say  at  P,  Q,  R.  It  is  plain  from  E~j-/ 
the  diagram,  that  the  perimeter 
of  the  polygon  formed  by  such 

tangents  will  approach  the  curve  more  and  more  closely 
as  the  number  of  the  tangents  is  increased.  Hence, 
when  the  number  becomes  infinite,  the  perimeter  will 
coincide  with  the  curve. 

Suppose  then  that  ^,  p,  v  being  continuous  variables  we 
connect  them  by  an  equation  expressing  the  condition 
that  every  line  represented  by  Xx  +  py  -f  v  =  0  shall 
touch  LM.  We  shall  then  have  an  infinite  system  of 
tangents  to  LM;  that  is,  we  shall  have  the  curve  itself. 
Hence,  the  equation  of  condition  in  /,  /jt,  v  may  be  taken 
as  the  symbol  of  the  curve,  and  is  called  its  tangential 
equation  for  obvious  reasons. 

247.  To  form  the  tangential  equation  to  any  curve, 
we  therefore  have  only  to  find  the  condition  in  ^,  //,  v 
which  must  be  fulfilled  in  order  that  the  line 

fa  +  py  +  v  =  0,  or  la  +  p$  -f  ^  =  0, 

may  touch  the  curve.  This  may  be  done  by  elimi- 
nating one  variable  between  the  equation  to  the  line  and 
that  of  the  curve,  and  then  forming  the  condition  that 
the  resulting  equation  may  have  equal  roots.  For  the 
roots  of  the  resulting  equation  are  the  co-ordinates 
of  the  points  in  which  the  line  cuts  the  curve  (Art.  62, 
Cor.  1) ;  and,  if  these  roots  are  equal,  the  points  of 


ENVELOPES.  227 

section  become  coincident  (that  is,  consecutive),  and  the 
line  is  a  tangent. 

Instead  of  this  method,  it  is  often  preferable  to  employ 
another,  which  will  be  illustrated  a  little  farther  on. 

248.  The  Right  Line  in  Tangentials. — This  is 
represented  not  by  an  equation,  but  by  co-ordinates.     The 
symbol  of  a  fixed  right  line  is  therefore 

a'  A  +  b'  fi  -f  c'  v  =  0 
a"X  +  V'fji  +  c"u  t=  0 

For,  by  solving  between  these  simultaneous  equations  to 
two  points,  we  can  fully  determine  the  ratios  X  :  /j.  :  v ; 
that  is,  we  can  determine  the  line  which  joins  the  points 
represented  by  the  given  equations. 

This  result  is  in  harmony  with  the  fact  already  no- 
ticed, that  the  Eight  Line  plays  in  the  tangential  system 
the  same  part  that  the  Point  does  in  the  Cartesian. 
In  the  abridged  notation,  the  simultaneous  equations 
a  =  0,  /9  —  0  are  the  tangential  symbol  of  a  right  line. 
For  the  sake  of  brevity,  we  shall  generally  speak  of  the 
line  joining  the  points  a  and  ft  (that  is,  the  points  whose 
equations  are  a  =  0,  /3  =  0)  as  the  line  «/9;  just  as,  in 
the  trilinear  system,  we  call  the  point  in  which  the  lines 
a  and  ft  intersect,  the  point  aft. 

ENVELOPES. 

249.  We  have  called  the  geometric  equivalent  of  a 
Cartesian   or   trilinear   equation   the   locus   of   a   point. 
Similarly,    the    geometric    equivalent    of    a    tangential 
equation   is   called   the   envelope   of  a  right  line.     This 
term  needs  explanation. 

Every  tangential  equation  takes  the  co-efficients  of 
la.  -f  ftp  -f-  v?  =  0  as  variables.  It  therefore  implies  the 


228  ANALYTIC  GEOMETRY. 

existence  of  an  infinite  series  of  right  lines,  the  successive 
members  of  which  have  directions  differing  by  less  than 
any  assignable  quantity,  and  intersections  lying  infinitely 
near  to  each  other.  Such  a  series  of  consecutive  direc- 
tions, blending  at  consecutive  points,  will  of  course  form 
a  curve.,  whose  figure  will  depend  on  the  equation  con- 
necting the  variable  co-efficients  /,  /j,  v.  A  curve  thus 
defined  by  a  right  line  whose  co-ordinates  vary  continu- 
ously, is  what  we  mean  by  an  envelope.  From  the  rea- 
soning of  Art.  246,  it  is  evident  that  a  right  line  always 
touches  its  envelope. 

Definition. — The  Envelope  of  a  right  line  is  the 
series  of  consecutive  directions  to  which  it  is  restricted 
by  given  conditions  of  form. 

Or,  The  envelope  of  a  right  line  is  the  curve  which 
it  always  touches. 

Remark. — We  thus  come  upon  an  essential  distinction  between 
the  tangential  system  and  the  Cartesian.  In  the  latter,  a  curve  is 
conceived  to  be  the  aggregate  of  an  infinite  number  of  positions; 
in  the  former,  it  is  regarded  as  the  complex  of  an  infinite  number 
of  directions.  It  appears  from  this  article,  that,  instead  of  calling 
the  condition  that  a  right  line  shall  touch  a  curve  the  tangential 
equation  to  the  curve,  we  may  say  it  is  the  equation  to  the  envelope 
of  the  line. 

25O.  Tangential  equation  to  OBC  Conic  cir- 
cumscribed about  a  Triangle. — We  might  form  this 
by  the  method  of  Art.  247,  but  can  proceed  more  rapidly 
as  follows :  —  The  trilinear  equation  to  the  tangent  of  a 
conic,  referred  to  the  inscribed  triangle,  (Art.  240,  II)  is 

la       mft      nr  m 

^72-f^+p=-  1), 

the  equation  to  the  conic  itself  being 


THE  CONIC  AS  ENVELOPE.  229 

Hence,  in  order  that  the  line  Xa  +  ///?  -\-  vy  =  0  may- 
touch  the  curve,  we  must  have  ),a!2  =  pl,  fjtpf2  =  pm, 
vf'2  =  pn;  that  is, 


But  a1  ',  /9',  /  are  the  co-ordinates  of  the  point  of  contact, 
and  must  satisfy  the  equation  to  the  curve.  Therefore, 
after  substituting  in  (2)  the  values  just  found, 

1/0  -f-  V'rnjjL  -f  l/m>  =  0 

is  the  condition  that  /«  -f  ^  -j-  ^  =  0  shall  touch  the 
conic.  In  other  words,  it  is  the  tangential  equation  to 
the  conic. 

Kemark.  —  By  clearing  this  equation  of  radicals,  we  shall  find 
that  it  is  of  the  second  degree  in  A,  ^  v. 

251.  The  further  investigation  of  tangentials  requires 
that  we  now  turn  aside  from  our  direct  path,  to  examine 
the  method  of  finding  the  envelope  of  a  right  line  when 
the  constants  which  enter  its  equation  are  subject  to 
given  conditions. 

By  the  terms  of  this  problem,  the  equation  to  the  line 
may  be  written  so  as  to  involve  a  single  indeterminate 
quantity  ;  for  instance,  in  the  form 

m2a  —  2kmf  -f  &/9  =  0, 

where  m  is  indeterminate,  and  k  is  fixed.  For,  by  means 
of  the  given  conditions,  we  can  eliminate  one  of  the  arbi- 
trary constants  which  enter  the  original  equation,  and 
thus  leave  but  one. 

Now,  by  the  definition  of  an  envelope  (Art.  249),  the 
right  line 

m2a  —  Zkm?  +  kfi  =  0  (1), 


230  ANALYTIC  GEOMETRY. 

is  tangent  to  its  envelope.  In  this  particular  instance, 
then,  the  envelope  is  such  a  curve  that  only  two  tangents 
can  be  drawn  to  it  from  a  given  point.  For  if  we  suppose 
the  line  (1)  to  pass  through  the  given  point  oJfi'f,  we 
shall  have 

a'.m2  —  Zkf.m  +  fc^  =  0  (2), 

from  which  to  find  m,  the  indeterminate  in  virtue  of 
which  (1)  represents  a  system  of  tangents  to  the  envelope; 
and,  since  (2)  is  a  quadratic  in  m,  only  two  lines  of  the 
system  can  pass  through  a'fi'f.  This  property  of  the 
envelope  is  designated  by  calling  it  a  curve  of  the  Second 
class :  the  class  of  a  curve  being  determined  by  the 
number  of  tangents  that  can  be  drawn  to  it  from  any 
one  point.  We  learn,  then,  that  if  the  equation  to  a  right 
line  involves  an  indeterminate  quantity  in  the  nth  degree, 
the  envelope  is  a  curve  of  the  nth  class.  * 

The  question  still  remains,  How  shall  we  obtain  the 
equation  to  this  envelope?  The  definition  of  Art.  249 
answers  this ;  for  since  the  envelope  is  the  series  of 
consecutive  directions  of  the  tangent,  we  have  only  to 
form  the  condition  that  the  n  values  of  the  indeterminate 
m  may  be  consecutive,  and  the  required  equation  is  found. 
Now  this  condition  is  of  course  the  same  as  that  which 
requires  the  equation  in  m  to  have  equal  roots.  For 
example,  the  equation  to  the  envelope  of  (1)  is  aft  =  kf2. 

Hence,  To  find  the  envelope  of  a  right  line,  throw  its 
equation,  by  means  of  the  given  conditions,  into  a  form 
involving  a  single  indeterminate  quantity  in  the  nth  degree : 
the  condition  that  this  equation  in  m  shall  have  equal  roots, 
is  the  equation  to  the  envelope. 


*  The  student  must  not  confound  the  class  of  a  curve  with  its  order. 
A  conic  belongs  to  the  Second  order,  and  also  to  the  Second  class;  hut 
other  curves  do  not  in  general  show  this  agreement. 


TANGENTIAL  EQUATION  TO  CONIC.  231 

Example.  —  Given  the  vertical  angle  and  the  sum  of  the  sides  of 
a  triangle  :  to  find  the  envelope  of  the  base. 

Take  the  sides  for  axes  ;  then  the  equation  to  the  base  is 


in  which  o  and  b  are  subject  to  the  condition  a  -f  b  —  c.  The  equation 
to  the  base  may  therefore  be  written 

«2  +  (y  —  *  —  <0  a  +  ex  =  0. 
Hence,  the  equation  to  the  envelope  is 

(y  —  *-c)2  =  4cx; 
or,  x'2  -  Ixy  +  j/2  —  2cx  —  2cy  +  c2  =  0. 

The  required  envelope  is  therefore  (Art.  191)  a  parabola;  which 
touches  the  sides  x=  0  and  y  —  0,  since  the  equations  for  deter- 
mining its  intercepts  on  the  axes  are 

x1  —  lex  -f  c2  =  0,     if  —  Icy  -f  c2  =  0. 

We  may  now  resume  the  direct  course  of  our  investi- 
gation, and  finish  this  part  of  it  by  establishing,  in  their 
proper  order,  the  theorems  necessary  for  interpreting 
tangential  equations  of  the  first  and  second  degrees. 

252.  Tangential  equation  to  any  Conic.  —  We 

shall  here  follow  the  method  of  Art.  247,  and  find  the 
condition  that  Xa.  -f-  ///9  -f-  wf  =  0  may  touch  the  conic 

Aa?  -f  B^  -f  Cf  +  2Ffr  +  2  GTOL  -f  ZHafi  =  0. 

Eliminating  f  between  the  equation  to  the  line  and  that 
of  the  conic,  and  collecting  the  terms  of  the  result  with 
reference  to  a  :  /9,  we  obtain 


?2=  0. 

Forming  the  condition  that  this  complete  quadratic  in 
a  :  /?  may  have  equal  roots,  we  get  the  required  tangential 
equation,  namely, 


—  Guv— 

An.  Ge.  23. 


232  ANALYTIC  GEOMETRY. 

which,  after  expansion  and  division  by  v2,  becomes 

~ 


Now  it  is  remarkable  that  the  co-efficients  of  (1) 
are  the  derived  polynomials  (see  Alg.,  411)  of  the 
Discriminant  J,  obtained  by  supposing  the  variable  to 
be  successively  A,  B,  C,  F,  6r,  H.  They  may  there- 
fore be  aptly  denoted  by  a,  6,  <?,  2/,  2g,  2k,  so  that  (1) 
shall  be  written 


atf  -f  bfjf  -f  cJ  +  2///V  +  2gvX  -f  2%  ^0       (2). 

Corollary.  —  If  we  represent  the  Discriminant  of  (2), 
namely, 

abc  -f  2h  —a2  —  b2  —  ch* 


by  d,  and  substitute  for  a,  6,  <?,  etc.,  their  values  from  (1), 
we  shall  obtain  the  relation 


a  property  which  has  some  important  bearings. 

353.  Theorem.  —  The  envelope  of  a  right  line  whose 
co-ordinates  are  connected  by  any  relation  of  the  first 
degree,  is  a  point. 

For  (Art.  243)  every  such  relation  is  the  tangential 
equation  to  a  point  through  which  the  line  always  passes 
(that  is,  to  which  it  is  always  tangent)  ;  and  (Art.  249, 
Rem.)  the  tangential  equation  is  the  equation  to  the 
envelope. 

Corollary.  —  By  means  of  the  relation  aA  -f  bfj.  -f-  cv  —  0, 
we  can  eliminate  either  A  :  v  or  /j.  :  v  from  the  equation 
Aa  -\-  fift  +  isf  —  0,  and  so  cause  it  to  involve  only  a 
single  indeterminate,  of  the  first  degree.  Hence,  If  the 


ENVELOPE  OF  SECOND  CLASS.  233 

equation  to  a  right  line  involves  a  single  indeterminate 
quantify  in  the  first  degree,  the  envelope  of  the  line  is 
a  point.  [Compare  Art.  116.] 

This  corollary  only  carries  out  the  principle  of  Art. 
251.  For  a  point  may  of  course  be  regarded  as  a 
curve  of  the  First  class. 

254.  Theorem.  —  The  envelope  of  a  right  line  whose 
co-ordinates  are  connected  by  any  relation  of  the  second 
degree,  is  a  conic. 

We  are  here  required  to  prove  that  any  tangential 
equation  of  the  form 

-f  cv2  +  2fp»  -f  2#vA  -f  ZUfjL  =  0, 


in  which  a,  b,  c,  /,  </,  h  are  any  six  constants  whatever, 
represents  a  conic.  Eliminating  v  between  the  given 
equation  and  la.  -f  pP  +  v?  —  0,  we  find 


(af  —  tyf-a.  -f  ca?)  A2  +  2  (hf—g$?  —fya  + 

-h(6r2-2//9r- 
Hence,  the  equation  to  the  envelope  of  the  line  is 


that  is,  after  expanding  and  dividing  through  by  f2, 
(S,-/2)  «'+  (ac- 


_ 


or,  since  the  co-efiicients  are  the  derived  polynomials 
of  3,  supposing  the  variable  to  be  successively  a,  b,  c, 
etc.,  and  may  therefore  be  represented  by  A,  B,  (7,  etc., 


?  -f  Cf  -f  2Ffr  +2Gra  +  2ffap  =0    (2)  : 

which  is  the  trilinear  equation  to  a  conic.     Our  propo- 
sition is  therefore  established. 


234  ANALYTIC  GEOMETRY. 

Corollary. — By  means  of  the  relation  given  in  the 
hypothesis  of  the  theorem  above,  we  can  cause  the 
equation  ha  -}-  fj.fi  -f-  vy  =  0  to  involve  but  a  single  inde- 
terminate, say  X  :  v,  in  the  second  degree.  Hence,  If 
the  equation  to  a  right  line  involves  an  indeterminate  quan- 
tity in  the  second  degree,  the  envelope  of  the  line  is  a  conic. 

This  conclusion  might  have  been  deduced  at  once 
from  the  equation  of  Art.  251,  namely, 

m2a  —  2mJcf  -f-  &/?  —  0. 

For  this  is  a  general  type  for  all  equations  answering 
the  description  of  the  corollary,  and  the  equation  to  the 
envelope  of  the  corresponding  line  is 

a^  =  Tcf, 

which,  as  we  have  seen  in  Ex.  1,  p.  221,  is  the  equation 
to  a  conic,  referred  to  two  tangents  and  their  chord  of 
contact. 

255.  Interchange  of  the  Trilinear  and  Tan- 
gential equations  to  a  Conic. — If  we  compare 
equation  (1)  of  the  preceding  article  with  equation  (1) 
of  Art.  252,  it  becomes  apparent  that  the  former  is 
derived  from 

a/?  -f-  b/j.2  +  cv2  +  2/Juv  -f  2gvl  +  Ship  =  0 

by  exactly  the  same  series  of  operations  by  which  the 
latter  is  derived  from 

Aa2  +  B^  -f  Of  +  2jP/9r  +  2  Gra  -f  Zffafi  =  0. 
Hence,  To  form  the  tangential  equation  to  a  conic  when 
its  trilinear  equation  is  given,  replace  a,  /9,  7-  by  X,  p,  v, 
and  the  co-efficients  A,  B,  C,  etc.,  by  the  corresponding 
derived  polynomials  of  d ;  and  to  form  the  trilinear 
equation  from  the  tangential,  replace  X,  p,  v  by  a,  /?,  f, 
and  the  co-efficients  a,  b,  c,  etc.,  by  the  derived  polynomials 
of  8. 


PRINCIPLE  OF  D  UALITY. 


235 


256.  It  is  obvious  that  the  application  of  the  fore- 
going principles  will  often  facilitate  the  investigation  of 
envelopes.  We  may  make  the  following  summary  of 
results : 

I.  Every   tangential    equation   of   the   first   degree 
represents  a  point. 

II.  Every  tangential  equation  of  the  second  degree 
represents  a  conic. 

III.  A  tangential  equation  of  the  wth  degree  represents 
a  curve  of  the  nth  class. 


RECIPROCAL    POLARS. 

257.  Reciprocal  relation  between  Points  and 
Uiies. — We  have  already  noticed  the  reciprocity  of  Car- 
tesian and  tangential  equations,  as  suggested  by  the  fact 
that  the  Point  and  the  Right  Line  interchange  their  offices 
in  passing  from  one  system  to  the  other.  This  remarkable 
property,  however,  does  not  appear  in  its  full  significance 
until  we  apply  to  tangentials  the  same  system  of  abridged 
notation  that  converts  a  Cartesian  into  a  trilinear  equa- 
tion. When  this  notation  is  applied,  it  is  found  that  an 
equation  in  «,  /2,  7-  or  u,  v,  w  is  susceptible  of  two  inter- 
pretations, according  as  it  is  read  in  trilinears  or  in  tan- 
gentials ;  and  gives  rise  to  two  distinct  theorems  (one 
relating  to  points,  the  other  to  lines),  which  in  view  of 
their  derivation  may  not  inaptly  be  styled  reciprocal 
theorems. 

This  capability  of  double  interpretation  is  known 
among  mathematicians  as  the  Principle  of  Duality,  and 
has  led  to  many  of  the  most  striking  results  of  the 
Modern  Geometry.  A  few  illustrations  will  enable  the 
student  to  conceive  of  the  principle  clearly. 


236  ANALYTIC   GEOMETRY. 

Suppose,  for  brevity,  we  write  $  =  0,  /$"  =  0  as  the  equations 
to  two  conies,  either  Cartesian  or  tangential.  Then  the  equation 
S-\-  kS'  =  Q,  being  satisfied  either  by  the  co-ordinates  of  points 
which  render  S  and  /S"  simultaneously  equal  to  zero,  or  by  the 
co-ordinates  of  lines  which  effect  the  same  result,  denotes  in  tri- 
linears  a  conic  which  passes  through  the  four  points  in  which  the 
conic  *S*  cuts  the  conic  $x,  and  in  tangentials  a  conic  which  touches 
the  four  lines  that  touch  S  and  S'  in  common. 

Similarly  ay  —  kfid,  being  the  equation  to  a  conic  since  it  is  of 
the  second  degree,  may  be  read  either  in  trilinears  or  tangentials. 
It  is  obviously  satisfied  by  either  of  the  four  conditions 


Hence,  in  trilinears  it  denotes  a  conic  passing  through  the  four 
points  a/3,  /?/,  7$,  <5a;  that  is,  circumscribed  about  the  quadrilateral 
whose  sides  are  the  four  lines  a,  /?,  7,  o:  while  in  tangentials  it 
represents  a  conic  touching  the  four  lines  aft,  (3y,  -yd,  6a;  that  is, 
inscribed  in  the  quadrilateral  whose  vertices  are  the  four  points 
«,  /?,  7,  6. 

Again,  £+  kafi  —  0  is  a  conic  passing  through  the  four  points 
in  which  the  lines  a  and  fi  cut  the  conic  S,  or  touching  the  four 
lines  drawn  from  the  points  a  and  J3  to  touch  the  conic  *S'  If,  then, 
wo  have  three  conies  S,  S+  kafi,  S-{-  k^ay,  ^e  may  either  say  that 
all  three  pass  through  the  two  points  in  which  the  line  a  cuts  S,  or 
that  all  three  touch  the  two  lines  drawn  from  the  point  a  to  touch  &. 

We  can  now  exemplify  the  method  of  obtaining  reciprocal  theo- 
rems. The  three  conies  S,  S  -}-  kafi,  S+k^a-y,  as  we  have  just 
shown,  all  pass  through  the  two  points  in  which  the  line  a  cuts 
them.  Moreover,  the  line  fi  evidently  joins  the  two  remaining 
points  in  which  S  cuts  *S'+  kafi;  the  line  7  joins  the  two  remaining 
points  in  which  $  cuts  S  +  k'ay  ;  while,  for  the  line  joining  the  two 
remaining  points  in  which  $-j-  kafi  cuts  S-\-  £'07,  we  get,  by  elim- 
inating between  these  equations,  kfi  —  k'y  =  Q.  Now  (Art.  10S) 
this  last  line  must  pass  through  the  point  fi-y.  Hence,  we  have  the 
following  theorem  : 

I.  If  three  conies  have  two  points  common  to  all,  the  three,  lines 
joining  the  remaining  points  common  to  each  two,  meet  in  one  point. 

Let  us  now  take  the  same  equations  in  tangentials.  The  two 
tangents  from  a  are  common  to  the  three  conies,  the  pair  from  (3  is 


RECIPROCAL  POLAES.  237 

common  to  the  first  and  second,  the  pair  from  7  is  common  to  the 
first  and  third,  while  the  pair  common  to  the  second  and  third 
intersect  in  the.  point  kfi  —  k'y.  But,  on  the  analogy  of  Art.  108, 
the  latter  point  is  on  the  line  /3y.  Hence  the  reciprocal  theorem: 

IT.  If  three  conies  have  two  tangents  common  to  all,  the  three  points 
in  which  the  remaining  'tangents  common  to  each  two  intersect,  lie  on 
one  right  line. 

By  comparing  the  phraseology  of  I  and  II,  we  see 
that  either  may  be  derived  from  the  other  by  simply 
interchanging  the  words  point  and  tangent,  and  point 
and  line.  In  fact,  if  the  reader  chooses  to  push  his 
inquiries  by  consulting  other  authors  upon  this  subject, 
he  will  find  that  the  entire  process  of  reciprocation,  as 
it  is  called,  may  be  reduced  to  the  operation  of  inter- 
changing the  terms  point  and  line,  chord  and  tangent, 
circumscribed  and  inscribed,  locus  and  envelope,  etc. 


o  Geometric  meaning  of  the  Reciprocal  Re- 
lation. —  The  process  of  reciprocation  being  so  mechan- 
ical, the  student  may  very  naturally  ask  how  we  can  be 
certain  that  reciprocal  theorems  are  any  thing  more 
than  fanciful  trifling  with  words.  As  a  sufficient 
answer  to  this  question,  we  shall  now  show  that  if  a 
given  theorem  is  proved  of  a  certain  curve,  we  can 
always  generate  a  second  curve  from  the  first,  to  which 
the  reciprocal  of  the  given  theorem  will  surely  apply. 
In  short,  we  shall  show  that  the  reciprocity  which  we 
have  illustrated  is  not  merely  a  property  of  trilinear  and 
tangential  equations  identical  in  form,  but  that  the  curves 
to  which  such  equations  belong  are  reciprocal. 

The  truth  of  this  statement  will  appear  in  two  steps: 
we  shall  first  explain  the  meaning  and  establish  the 
existence  of  reciprocal  curves;  and  then  prove  that  the 
tangential  equation  to  a  curve  is  the  trilinear  equation 
to  its  reciprocal. 


238  ANALYTIC   GEOMETRY. 


259.  Generation    of  Reciprocal    CnrYes.  —  To 

explain  this,  and  establish  its  possibility,  we  shall  have 
to  anticipate  a  single  property  of  conies.  The  theorem 
will  be  proved  in  its  proper  place  in  Part  II,  but  for  the 
present  the  student  must  take  it  upon  trust. 

Every  conic,  then,  is  characterized  by  the  following 
twofold  property : 

I.  If  different  chords  to  a  conic  be  drawn  through  the 
same  point,  and  tangents  to  the  curve  be  formed  at  the 
extremities  of  each  chord,  the  intersections  of  all  these  pairs 
of  tangents  ivitt  lie  on  the  same  rigid  line. 

II.  If  different  pairs  of  tangents  be  drawn  to  a  conic 
from  points  lying  on  the  same  right  line,  and  chords  be 
formed  joining  the  points  of  contact  belonging  to  each  pair, 
all  these  chords  of  contact  will  intersect  in  the  same  point. 

From  this  it  appears,  that,  in  relation  to  any  conic, 
there  is  a  certain  right  line  determined  by,  and  there- 
fore corresponding  to,  any  assumed  point ;  and  a  certain 
point  determined  by,  and  therefore  corresponding  to, 
any  assumed  right  line.  This  interdependence  of  points 
and  lines  is  expressed  by  calling  the  point  the  pole  of 
the  line,  and  the  line  the  polar  of  the  point. 

If  the  student  will  now  draw  diagrams,  forming  the 
polar  of  a  point  according  to  I,  and  the  pole  of  a  line 
according  to  II,  he  will  find  that  when  a  point  is  within 
the  conic  (a  circle  will  be  most  convenient  for  illustration), 
its  polar  is  without;  that  when  the  point  is  without  the 
conic,  its  polar  is  within,  and  in  fact  is  the  chord  of 
contact  of  the  two  tangents  drawn  from  the  point ;  that 
when  the  point  is  on  the  conic,  its  polar  is  also  on  the 
curve  —  in  fact,  is  the  tangent  at  the  point.  Conversely, 
if  a  right  line  is  without  a  conic,  its  pole  is  within ;  if  the 
line  is  within  (that  is,  if  it  forms  a  chord),  its  pole  is 


RECIPROCAL  CURVES. 


239 


without,    and   is   the   intersection   of   the   two   tangents 

drawn  at  the  extremities  of  the  chord ;  if  the  line  is  on 

the  curve,  the  pole  is  the  point  of  contact.     Thus,  in  the 

diagram,  P  is  the  pole  of  LM9 

and  LM  the  polar  of  P;  L  is 

the  pole  of  T'T,  and  T'Tthe 

polar  of  L;  M  is  the  rjole  of 

V  F,  and  V  Fthe  polar  of  If; 

T  is  the  pole  of  LT,  and  LT 

the  polar  of  T ';   and  so  on. 

The  pole  is  said  to  correspond 

to  its  polar,  and  reciprocally. 

It  is  obvious  that  as  the  polar  changes  its  position, 
the  position  of  the  pole  is  changed ;  so  that,  if  the  polar 
determine  a  curve  as  its  envelope,  the  pole  will  deter- 
mine another  as  its  locus.  Suppose,  then,  we  have  any 
curve  S9  and  relate  it  to  the  conic  2  by  taking  the  pole 
with  respect  to  2  of  any  tangent  to  S:  the  locus  of  the 
pole  will  be  a  second  curve  s,  which  may  be  called  the 
polar  curve  of  S.  It  is  evident  that  every  point  of  s 
will  correspond  to  (that  is,  be  the  pole  of)  some  tangent 
to  S.  Therefore,  if  we  take  any  two  points  on  s,  they 
will  at  the  same  time  determine  a  chord  of  s,  and  the 
intersection  of  two  tangents  to  S;  that  is,  every  chord 
of  s  is  the  polar  of  the  intersection  of  the  two  tangents 
to  S  which  are  the  polars  of  the  extremities  of  that 
chord.  Hence,  supposing  the  points  of  s  to  be  consecu- 
tive, and  the  corresponding  tangents  to  S  on  that  account 
to  intersect  on  the  curve,  we  have :  Every  point  of  S  is 
the  pole  with  respect  to  2'  of  some  tangent  to  s.  That  is, 
as  s  is  the  locus  of  the  pole  of  any  tangent  to  S9  so  $  is 
the  locus  of  the  pole  of  any  tangent  to  s.  Or,  in  other 
words,  a  curve  and  its  polar  with  respect  to  a  fixed  conic 
2  may  be  generated  each  from  the  other  in  exactly  the 


240  ANALYTIC  GEOMETRY. 

same  manner.     A  curve  and  its  polar  are  thus  seen  to 
be  reciprocal  forms. 

Given,  then,  any  curve  whatever,  by  means  of  a  fixed 
conic  —  we  can  always  generate  a  second  curve,  which 
may  properly  be  called  the  reciprocal  polar  of  the  first. 

260.  The  tangential  equation  to  a  Curve,  the 
trilinear  equation  to  its  Reciprocal. — This  theo- 
rem is  clearly  true  ;  for  the  co-ordinates  of  any  tangent 
to  S  may  of  course  be  taken  as  the  co-ordinates  of  its 
pole,   that  is,   as  the   co-ordinates   of  any  point   on  s: 
hence,  the  equation  to  the  envelope  $  may  be  regarded 
as  the  equation  to  the  locus  s. 

261.  Since  it  thus  appears  that  every  curve  has  its 
reciprocal,  whose  equation  is  identical  with  the  tangential 
equation  to  the  curve,  it  follows  that  all  the  results  obtain- 
able either  by  mechanical  reciprocation  or  by  the  double 
interpretation  of  equations,  are  real  properties  of  real 
curves.     Given,  then,  any  equation  in  «,  ^9,  7%  we  are  not 
only  justified  in  reading  it  both  in  trilinears  and  in  tan- 
gentials,  but  must  so  read  it,  if  we  wish  to  exhaust  its 
geometric  meaning. 

Note — From  the  relation  now  established  between  the  curves 
corresponding  to  the  two  interpretations  of  an  equation  in  a,  /?,  y, 
the  method  of  deriving  reciprocal  theorems  is  sometimes  called 
the  Method  of  Reciprocal  Polars,  instead  of  the  Principle  of  Duality. 
It  should  be  stated,  however,  that  both  these  terms,  as  now  applied 
to  processes  purely  analytical,  are  borrowed  from  the  cognate  branch 
of  pure  geometry.  They  both  entered  the  history  of  Geometry  ;is 
titles  of  purely  geometric  processes,  and  the  larger  part  of  their 
remarkable  results  were  established  by  the  aid  of  the  diagram 
alone.  From  the  very  process  of  generating  a  reciprocal  polar,  it 
is  evident  that  the  Method  of  Reciprocals  contains  in  itself  the 
evidence  for  the  truth  of  all  theorems  based  upon  it,  and  need  not 
invite  the  aid  of  analysis. 


PROPERTIES  OF  RECIPROCALS.  241 

The  Principle  of  Duality,  as  a  purely  geometric  method,  is  duo 
to  the  French  mathematician  GKIKJOXXK;  its  first  presentation  in 
the  analytic  form  of  an  equation  with  double  meaning,  was  made 
by  PLUCkEE,  in  his  System  der  analytischen  Geometric,  1835.  The 
geometric  Method  of  Reciprocal  Polars  was  the  invention  of  PON- 
CELET,  who  presented  an  account  of  its  elements  in  Gergonne's 
Annules  de  MatMmatiqucs,  torn.  VIII,  1818;  and,  afterward,  an 
extended  development  of  its  general  theory  in  Crelle's  Journal  filr 
il/c  re  hie  und  angewandte  Mathcmatik,  Bd.  IV,  1829.  The  latter, 
however,  was  previously  read  in  1824  to  the  Royal  Academy  of 
Sciences  at  Paris,  and  led  to  a  dispute  between  Poncelet  and  Ger- 
gonne  as  to  the  prior  claims  of  the  Principle  of  Duality.  For  the 
discussion  which  ensued,  the  reader  is  referred  to  the  Annales, 
torn.  XVIII 

The  conic  JT,  upon  which  the  Principle  of  Duality  and 
the  Method  of  Reciprocal  Polars  as  analytic  processes 
are  based,  is  called  the  auxiliary  conic.  It  may  be  any 
fixed  conic  whatever,  but  is  in  practice  usually  a  circle; 
because  that  curve  enjoys  certain  properties  by  means 
of  which  we  can  reciprocate  theorems  concerning  magni- 
tude as  wTell  as  those  concerning  position.  The  use  of  a 
parabola  for  —  has  been  introduced  by  CHASLES,  but  the 
applications  of  which  his  method  is  capable  are  compara- 
tively few. 

We  shall  now  demonstrate  two  or  three  of  the  leading 
properties  of  reciprocals. 

262.  Theorem.  —  The  reciprocal  of  a  rigid  line  with 
rcxpi'ct  to  ~  is  a  point;  and  conversely. 

For  the  tangential  equation  of  the  first  degree  denotes 
a  point. 


Theorem.  —  TJie  reciprocal  of  a  conic  with  respect 
to  -  is  a  conic. 

For  the  trilinear  equation  to  a  conic  is  of  the  second 
degree,  and  every  equation  of  the  second  degree,  when 
interpreted  in  tangentials,  denotes  a  conic. 


242  ANALYTIC  GEOMETRY. 

264o  Theorem. — The  reciprocal  of  a  curve  of  the  nth 
order,  is  a  curve  of  the  nth  class. 

For  the  trilinear  equation  of  the  wth  degree,  when 
interpreted  in  tangentials,  denotes  a  curve  to  which  n 
tangents  can  be  drawn  from  a  given  point. 

265.  We  add  a  few  exercises  upon  various  subjects 
treated  in  this  Section,  premising  that  the  student  must 
not  suppose  them  to  be  adequate  illustrations  of  the  scope 
of  the  tangential  method  :  they  are  in  fact  only  useful  for 
fixing  the  leading  points  of  the  preceding  sketch.  The 
reader  who  wishes  to  see  the  very  remarkable  results 
of  the  principles  of  Duality  and  Reciprocal  Polars  may 
consult,  in  addition  to  the  works  already  mentioned, 
Chasles's  Traite  de  Geometric  Superieure,  Steiner's 
Entwickelung  der  Abhiingigkeit  geometrischen  Gestalten, 
etc.,  Booth's  treatise  On  the  Application  of  a  New 
Analytic  Method  to  the  Theory  of  Curves  and  Curved 
Surfaces,  Salmon's  Higher  Plane  Curves  and  Geometry 
of  Three  Dimensions,  and  Poncelet's  Traite  des  Proprietes 
Projectives. 

EXAMPLES. 

.1.  Interpret  in  tangentials  the  several  equations  of  the  example 
in  Art.  219. 

2.  Write  the  equations  to  the  points  (5,  6),  (—3,  2),  (7,  8,  —9). 

3.  What  is  the  tangential  symbol  of  the  right  line  passing  through 
(2,  3)  and  (4,5)? 

4.  Form  the  tangential  equation  to  the  circle  represented  by 

sin  A   |  sin  B      sin  C 

-^-  +  — +  —  -°- 

5.  Interpret  the  equation  aft  =  £y2  both  in  trilinears  and  tan- 
gentials. 

6.  Show  that  m2a  —  Imky  +  kj3  =  0  is  the  tangential  equation  to 
any  point  on  the  curve  a/3  =  ky2. 


EXAMPLES  IN  TANGENTIALS.  243 

7.  Find  the  equation  to  the  reciprocal  of  the  conic  represented 

*>y 

Via-}-  l/mj3  +  Vn-y  =  Q. 

8.  Find  the  equation  to  the  envelope  of  the  right  line  whose 
co-efficients  fulfill  the  condition 


and  the  equation  to  the  envelope  of  one  whose  co-efficients  satisfy 

V  A  +  l/m/T+  Vm>  =  0. 
What  is  the  meaning  of  the  results  ? 

9.  Prove  that  the  envelope  of  the  conic  represented  by 


in  which  I,  m,  n  are  subject  to  the  condition 

I/A?  -f  V~pn+  Vvn  =  0, 
is  the  right  line  Aa  +  ftp  -f-  vj  =  0. 

10.  Prove  that  the  envelope  of  the  conic  Vla-\-  Vmfi  +  l/ny  =  0, 
whose  co-efficients  satisfy  the  relation 


is  the  right  line  Aa  +  pft  +  VJ  =  0. 

11.  Find  the  envelope  of  a  right  line,  the  perpendiculars  to  which 
from  two  given  points  contain  a  constant  rectangle. 

12.  The  vertex  of  a  given  angle  moves  along  a  fixed  right  line 
while  one  side  passes  through  a  fixed  point:    to  find  the  envelope 
of  the  other  side. 

13.  A  triangle  is  inscribed  in  a  conic,  and  its  two  sides  pass 
through  fixed  points  :  to  find  the  envelope  of  its  base. 

14.  Prove  that  if  three  conies  have  two  points  common  to  all, 
their  three  reciprocals  will  have  two  tangents  common  to  all  ;  and 
conversely. 


244 


ANALYTIC   GEOMETRY. 


15.  Establish  the  following  reciprocal  theorems,  and  determine 
the  conditions  for  the  cases  noticed  under  them : 


If  two  vertices  of  a  triangle 
move  along  fixed  right  lines  while 
the  three  sides  pass  each  through 
a  fixed  point,  the  locus  of  the 
third  vertex  is  a  conic. 

But  if  the  points  through  which 
the  sides  pass  lie  on  one  right  line, 
the  locus  will  be  a  right  line. 

In  what  other  case  will  the 
locus  be  a  right  line? 


If  two  sides  of  a  triangle  pass 
through  fixed  points  while  the  three 
vertices  move  each  along  a  fixed 
right  line,  the  envelope  of  the 
third  side  is  a  conic. 

But  if  the  lines  on  which  the 
vertices  move  meet  in  one  point, 
the  envelope  will  be  a  point. 
[Compare  Ex.  39,  p.  125.] 

In  what  other  case  will  the 
envelope  be  a  point?  [Compare 
Ex.  40,  p.  125.] 


PLANE  CO-ORDINATES. 


PART  II. 

THE  PROPERTIES  OF  CONICS. 

266.  The  investigations  of  Part  I,  have  taught  us 
the  methods  of  representing  geometric  forms  by  analytic 
symbols ;  and  furnished  us,  in  the  resulting  equations  to 
curves  of  the  First  and  Second  orders,  with  the  necessary 
instruments  for  discussing  those  curves.  We  now  proceed 
to  apply  our  instruments  to  the  determination  of  the 
properties  of  the  several  conies.  We  shall  adhere  to 
the  order  of  Part  I,  considering  first  the  several  vari- 
eties in  succession,  and  then,  by  way  of  illustrating  the 
method  of  investigating  the  common  properties  of  a 
whole  order  of  curves,  determining  those  of  the  Conic 
in  General. 


CHAPTER   FIRST. 
THE  RIGHT  LINE. 

267.  Under  this  head,  we  only  purpose  developing 
a  few  properties,  noticeable  either  on  account  of  their 
usefulness  or  their  relations  to  the  new  or  to  the  higher 
geometry. 

(245) 


246 


ANALYTIC  GEOMETRY. 


268.  Area  of  a  Triangle  in  terms  of  its  Ver- 
tices.—  Let  the  vertices  be 
x\y\->  X2t/2i  x£)z->  represented  in 
the  diagram  by  A,  -B,  C.  It 
is  obvious  that  for  the  re- 
quired area  we  have 


A  BC=ALMB+BMNC  —  CALN  ; 
that  is, 


or 


OTT 


3T 


N      X 


2  7=  y,  (x-x?)  +  y2  (x-x,)  +  7/3  (x-x.2). 


Remark.  —  This  expression  for  the  double  area  of  the 
triangle  is  identical  with  that  which  in  Art.  112  is 
equated  to  zero  as  the  condition  that  three  points  may 
lie  on  one  right  line.  We  thus  discover  the  latter  con- 
dition to  be  simply  the  algebraic  statement,  that,  when 
three  points  lie  on  one  right  line,  the  triangle  which  they 
determine  vanishes:  which  obviously  accords  with  the 
fact. 

269.  Area  of  a  Triangle  in.  terms  of  its  inclos- 
ing lines.  —  Let  the  three  lines  be  Ax  -j-  By  -)-  C=  0, 

MX  +  B'y  +  C'  =  0,  A"x  +  B"y  +  C"  =  0.  Their  in- 
tersections will  form  the  vertices  of  the  triangle  ;  hence, 
substituting  for  x}y},  x2y2,  xzy^  in  the  preceding  formula, 
according  to  Art.  106,  we  obtain 


2T=  - 


C  A/  -  C/  A 


Y  C"  —  B" 


B"C    -B    C' 


A 

B/ 

—  A/B     i 

i  A/  B" 

—  A//B/        A"B    —A   B" 

C/ 

A" 

-  C 

//A/  ' 

\B"C 

-B 

C" 

B 

C'  — 

7D/     /-^ 

A/ 

B" 

-A 

"B' 

(A"B 

A 

B" 

A 

B/    - 

A/  B 

C"A 

Q 

A"  i 

\B   C' 

—  w 

C 

B/ 

C"  — 

B//C/ 

A"B    —A  B"(A   B/  —A/B 


AREA  OF  A  TRIANGLE.  247 

Now  if  we  reduce  each  of  the  sets  of  fractions  inside  the 
braces  to  a  common  denominator,  the  three  new  numera- 
tors will  be  respectively 

E"  I  B(  C"A'—  C/  A  ")  +B'(  CA"-C"  A)  +B"(  C'  A—CA')  \  , 
B  \B(C''A'—C/A"}+B'(CA"-C"A}+B"(C/A-CA'}\, 
B'  lB(C/'A/—C/A//)+B'(CA"-C//A)+B"(C/A—CA')l. 

Hence,  the  final  expression  for  the  required  area  may  be 
written 

I  B(  C"A'—  CM") 


(AB'-A'B)  (A'B"-A"B')(AB"—A"B) 

Remark,  —  The  numerator  of  this  expression  may  be 
otherwise  written 


so  that  (Art.  113),  if  the  three  lines  pass  through  the 
same  point,  2  T  vanishes.  On  the  other  hand,  the 
expression  for  2T  becomes  infinite  (Art.  96,  Cor.  2) 
whenever  any  two  of  the  lines  are  parallel.  In  both 
respects,  then,  the  formula  accords  with  the  facts. 

27O.  Ratio   in   which    the   instance    between 
Two  Points   is   divided   by    a    Given   Line.  —  Let 

m  :  n  be  the  ratio  sought,  and  x\y^  x.2y2  the  given  points. 
The  co-ordinates  of  division  (Art.  52)  will  then  be 

^  ___  mx2  +  nxl  __  my,  -H?|/, 

m  +  n  m  -f-  n 

But  these  must  of  course  satisfy  the  equation  to  the  given 
line  ;  hence, 

A  (mx.2  -f  nxj  -f-  B  (my,  -f-  ny,}  +  C(m  +  ri)  =  0. 

m_       Ax,  +       i  +  0 
n~ 
An.  Ge.  24. 


248  ANALYTIC  GEOMETRY. 

Corollary.  —  If  the  given  line  passed  through  two  fixed 
points  x3y3  and  x4y4,  we  should  have  (Art.  95,  Cor.  1) 


_ 
n~        (y-,  —  y,)  x2  —  (xz  —  x,)  y2 

as  the  ratio  in  which  the  distance  between  two  fixed  points 
is  divided  by  the  line  joining  two  others. 

TRANSVERSALS. 

27  JU   Definition,  —  A  Transversal  of  any  system  of 
lines  is  a  line  which  crosses  all 
the  members  of  the  system. 

Thus,  LMN  is  a  transversal 
of  the  three  sides  AB,  BC,  CA 
in  the  triangle  ABC. 


.  Theorem.  —  In  any  triangle,  the  compound  ratio 
of  the  segments  cut  off  upon  the  three  sides  by  any  trans- 
versal is  equal  to  —  1. 

In  the  above  diagram,  let  the  vertices  A,  B,  C  be 
represented  by  #,?/,,  x.2y2,  x3y?t,  and  the  transversal  LN 
by  Ax  +  By  -f-  C=  0.  Then  (Art.  270) 


AL    LB  =  -  (Axl+Byl+  C)  :  (Ax2+By2+  C), 
BM:MC=  — 

CN  :  NA  =  — 


Multiplying   these    equations    member   by    member,   we 
obtain  the  proposition. 

273.  Theorem.  —  In  any  triangle,  the  compound  ratio 
of  the  segments  cut  off  upon  the  three  sides  by  any  three 
convergents  that  pass  through  the  vertices  is  equal  to  -f-  1. 

For,  if  the  point  of  convergency  be  0,  represented 
by  x4y4  while  the  vertices  are  denoted  as  in  the  preceding 


THREE  CONVERGENTS. 


249 


article,  we  shall  have  (Art.  270,  Cor.),  after  merely  re- 
arranging the  terms, 


PB 


_K»(yi—  yi)+si(ys—  y^+^Cyi—  2/2) 


and   the   product   of   these    equations   is   the   algebraic 
expression  of  the  theorem. 

We  shall  next  give  some  illustrations  of  the  uses  of 
abridged  notation,  as  applied  to  rectilinear  figures  and 
to  right  lines  in  general. 


TRIPLE    CONVERGENTS   IN   A    TRIANGLE. 

274.  Theorem.  —  The   three   bisectors    of  the   internal 
angles  of  a  triangle  meet  in  one  point. 

For  their  equations  are  a  —  /3=0,  ft  —  f—  0,  f  —  «—  0; 
and  (Art.  114)  these  vanish  identically  when  added 
together. 

275.  Theorem.  —  The    bisectors    of   any   two    external 
angles  of  a   triangle,  and  the   bisector  of  the  remaining 
internal  angle,  meet  in  one  point. 

For  their  equations  are  a-{-ft=Q9  j3-\-r=Q,  f  —  a=0, 
etc.  But  (Art.  108)  f  —  a  is  a  line  passing  through  the 
intersection  of  a-f-/?  and  /9-f  f. 

27G.  Theorem.  —  The  three  lines  which  join  the  vertices 
of  a  triangle  to  the  middle  points  of  the  opposite  sides  meet 
in  one  point. 

For  (Ex.  2,  p.  221)  their  equations  are  respectively 
asm  A  —  ft  sin  ,5—  0,  ftsinB  —  7-  sin  (7=0,  fsin(7—  asinyl=0, 
and  therefore  vanish  identically  when  added  together. 


250  ANALYTIC  GEOMETRY. 

2*7*7.  Theorem, — The  three  perpendiculars  let  fall  from 
the  vertices  of  a  triangle  upon  the  opposite  sides  meet  in 
one  point. 

For  (Ex.  3,  p.  221)  the  corresponding  equations  are 

a  cos  A — /9  cosB=Q,  ft  cosB — ?-  cos  (7=0, y  cos  C — a  cosJ.— 0. 

278.  Theorem, — The  three  perpendiculars  erected  at  the 
middle  points  of  the  sides  of  a  triangle  meet  in  one  point. 

For  (Ex.  4,  p.  221)  we  have  found  their  equations 
to  be 

a  sin  A  —  (3  sin  B  +  7  sin  (A— B)  =  0, 
psinB  —  ysin  C-f  a  sin  (B—  C]  =  0, 
7  sin  (7  —  asin^L-f  (3s'm(C—  A)=0; 

and,  if  we  multiply  these  by  sin2(7,  shrJ.,  sin2_B  respect- 
ively, we  shall  cause  them  to  vanish  identically.  (See 
Trig.,  850,  Ex.  2.) 

HOMOLOGOUS   TRIANGLES. 

2*70.  Definitions. — Two  triangles,  the  intersections  of 
whose  sides  taken  two  and  two  lie  on  one  right  line,  are 
said  to  be  homologous. 

The  line  on  which  the 
three  intersections  lie  is 
called  the  axis  of  homology. 
Any  two  sides  that  form 
one  of  the  three  intersec- 
tions are  termed  corre- 
sponding sides;  and  the 
angles  opposite  to  them, 
corresponding  angles. 

Thus,  in  the  diagram,  the  triangles  ABC,  AfB'Cf  are 
homologous  with  respect  to  the  axis  LMN.  AB,  A'B'; 
BC,  B'C' ;  CA,  C'A'  are  the  corresponding  sides;  and 
Ay  A';  B)  B';  (7,  C",  the  corresponding  angles. 


HOMOLOGY.  251 

280.  Theorem. — In  any  two  homologous  triangles,  the 
right  lines  joining  the  corresponding  vertices  meet  in  one 
point. 

Let  a,  /?,  f  be  the  sides  of  ABC,  and  take  the  latter 
for  the  triangle  of  reference ;  the  equation  to  the  axis 
LN  may  then  be  written  la  -f-  mft  -j-  ny  =  0.  Suppose 
the  Cartesian  origin  to  be  somewhere  between  the  tri- 
angles, and  (Art.  108,  Cor.  2)  the  equations  to  AB', 
B'C',  C'A,  which  pass  through  the  intersections  of 
a,  /?,  f  with  the  axis,  will  be 

(I—  Oa+m/?+n7=0,     la+(m—m/)/3+n-y=Q,     la+mp+(n— n')y=0. 

Subtracting  the  second  of  these  from  the  first,  the  third 
from  the  second,  and  the  first  from  the  third,  we  get 

I' a  —  m'p  =  0,     mfp  —  n'r  =  0,     n'r  —  Va  =  0. 

But  these  equations  (Art.  107)  evidently  denote  the  lines 
BB',  CO',  AA';  and  they  vanish  identically,  when  added 
together. 

Remark. — The  point  in  which  the  lines  joining  the 
corresponding  vertices  meet,  is  called  the  center  of 
homology. 

281.  Theorem. — If  the  lines  joining  the  corresponding 
vertices  of  two  triangles  meet  in  one  point,  the  intersections 
of  the  corresponding  sides  lie  on  one  right  line. 

This  theorem,  the  converse  of  the  preceding,  is  ob- 
tained at  once  by  merely  interpreting  the  equations  of 
the  foregoing  article  in  tangentials.  We  leave  the  student 
to  carry  out  the  details. 

COMPLETE    QUADRILATERALS. 

S82.  Definitions. — A  Complete  Quadrilateral  is 

the  figure  formed  by  any  four  right  lines  intersecting  in 
six  points. 


252  ANALYTIC  GEOMETRY. 

The  three  remaining  right  lines  by 
which  the  six  points  of  intersection 
can  be  joined  two  and  two,  are  called 
the  diagonals  of  the  quadrilateral. 

Thus,  ABCDEF  is  the  complete 
quadrilateral  of  the  four  lines  AB, 
BE,  EF,  FA,  which  meet  in  the  six 
points  B,  E,  F,  A,  C,  D.  AE,  BF  are 
the  two  inner  diagonals,  and  CD  is  the  outer  one,  some- 
times called  the  third. 

383.  Theorem.  —  In  any  complete  quadrilateral,  the 
middle  points  of  the  three  diagonals  lie  on  one  right  line. 

Let  a,  /?,  f,  d  be  the  equations  to  the  four  sides  of  the 
quadrilateral,  and  let  the  respective  lengths  of  BE,  EF, 
FA,  AB  equal  a,  b,  c,  d.  Then  L,  M,  N  being  the  middle 
points  of  the  three  diagonals,  we  have  the  following  equa- 
tions to  the  lines  drawn  from  the  vertices  to  the  middle 
points  of  the  bases  of  the  triangles  ABE,  EFA,  ABF, 
FEB: 

dd  —  aa  =  0  (BL),     bp  —  cr  =  0  (FL)  ; 
cr—dd=Q  (AM),    aa  -  6/9  =  0  (EM). 

Hence,  L  and  M  both  lie  upon  the  line 

aa  —  bp  +  cr  —  dd  =  Q  (1)  , 

as  this  obviously  passes  through  the  intersection  of 
(BL,  FL),  and  of  (AM,  EM).  If  we  now  put  Q  =  the 
double  area  of  ABEF,  we  shall  have 


and  thence 

aa  —  bp  +  cr—d8    =   =    2  (aa  +  cr)  — 


Q. 


AN  HARMONICS.  253 

Therefore  (Art.  229)  the  line  (1)  is  parallel  to  the  two 
lines  aa  -j-  cf  and  bp  -j-  dd,  and  midway  between  them. 
It  accordingly  bisects  the  distance  between  ;-«  (which  is 
a  point  on  the  first)  and  fld  (which  is  a  point  on  the 
second).  That  is,  N  lies  on  (1) :  which  proves  our 
proposition.  * 

THE   ANHARMONIC   RATIO. 

284.  Definition. — A  Linear  Pencil  is  a  group  of 
four  right  lines  radiating   from   one 
point. 

Thus,  OA,  OQ,  OB,  OP  constitute 
a  linear  pencil. 

2S5«  Theorem, — If  a  linear  pencil 
is  cut  by  any  transversal  in  four  points 
A,  Q,  B,  P,  the  ratio  AP.QB  :  AQ.PB  is  constant. 

For,  putting  p  =  the  perpendicular  from  0  upon  the 
transversal,  we  have  p.AP=  0 A. OP  sin  AOP;  p.QB  — 
OB.OQsmQOB;  p.AQ  =  OA.OQsinAOQ;  &ndp.PB  = 
OB.OPsiuPOB:  whence 

p\AP.  QB  =  OA. OB. OP.OQsmAOPsm  QOB, 
p\ AQ.PB  =  OA. OB. OP. OQsmAOQ sin  POB. 

AP.  QB  _  sin  A  OP  sin  QOB 
'  AQ.PB  ~~  sin  AOQ  sin  POB' 

a  value  which  is  independent  of  the  position  of  the 
transversal. 

Remark, — The  constant  ratio  just  established  is  called 
the  anharmonic  ratio  of  the  pencil.  By  reasoning  sim- 
ilar to  that  just  used,  it  may  be  shown  that  the  ratios 
AP.QB  :  AB.QP&nd  AB .QP  :  AQ.BP  are  also  con- 


*See  Salmon's  Conic  Sections,  Ex.  3,  p.  64. 


254  ANALYTIC  GEOMETRY. 

stant.  To  these,  accordingly,  the  term  anharmonic  is  at 
times  applied ;  but  it  is  generally  reserved  for  the  par- 
ticular ratio  to  which  we  have  assigned  it,  and  we  shall 
always  intend  that  ratio  when  we  use  it  hereafter. 

Note — The  Anharmonic  Ratio  has  an  important  place  in  the 
Modern  Geometry,  especially  in  connection  with  the  doctrine  of  the 
Conic.  Its  existence,  however,  has  been  known  since  the  time  of 
the  Alexandrian  geometer  PAPPUS,  who  gives  the  property  in  his 
Mathematics  Collectiones,  Book  VII,  129,  and  who  probably  belongs 
to  the  fourth  century.  The  name  anharmonic  was  given  by  CHASLES. 
But  the  bearing  of  the  ratio  upon  the  new  geometry  had  been  pre- 
viously investigated  by  MOBIUS,  who  called  it  the  Ratio  of  Double 
Section  (Doppelschnittsverhaltniss). 

286.  Definition. — An  Harmonic  Proportion  sub- 
sists between  three  quantities,  when  the  first  is  to  the 
third  as  the  difference  between  the  first  and  second  is  to 
the  difference  between  the  second  and  third. 

Thus,  if  the  pencil  in  the  above  diagram  cut  the  trans- 
versal so  as  to  make  AP:AQ  ::  AP—AB  :  AB  —  AQ, 
the  whole  line  AP  would  be  in  harmonic  proportion  with 
its  segments  AB  and  AQ. 

Corollary. — A  line  is  divided  harmonically,  when  it  is 
cut  into  three  segments  such  that  the  whole  is  to  either 
extreme  as  the  other  extreme  is  to  the  mean.     For  the. 
proportion  given  above  may  obviously  be  written 

AP:AQ::BP:BQ. 

287.  Definitions. — An   Harmonic   Pencil    is    one 

which  cuts  its  transversals  harmonically. 

I.  From  the  final  equation  of  Art.  285,  it  is  evident 
that  when  the  anharmonic  ratio  of  a  pencil  is  numerically 
equal  to  1,  the  pencil  is  harmonic. 

II.  The  four  points  in  which  a  line  is  cut  by  an  har- 
monic pencil,  are  called  harmonic  points. 


HARMONIC  PENCILS. 


255 


III.  Linear  pencils  are  in  general  termed  anharmonic, 
because  they  do  not  in  general  cut  off  harmonic  segments 
from  their  transversals. 

288.  Theorem.  —  The  anharmonic  of  the  pencil  formed 
by  the  four  lines  a,  ft  a  +  &ft  «  +  W$  is  equal  to  k  :  k'. 

Let  OA  be  the  position  of  #, 
OB  of  ft  OP  of  a  -f  &ft  and  0$  of 
a  -f-  &'ft  Then,  &  being  the  ratio  of 
the  perpendiculars  from  OP  upon 
0J.,  0.Z?,  and  Jcf  the  ratio  of  those 
from  OQ  on  the  same  lines,  we  shall 
have  &=sin^l0P:  siuPOB,  and  k'=amAOQ  :  smQOB. 

Hence>  k_  ,  siuAOPsiuQOB  t 

k'~  smAOQsinPOB  '' 

which  (Art.  285)  proves  the  proposition. 

Corollary.  —  If  kr  =  —  k,  the  anharmonic  of  the  pencil 
becomes  numerically  equal  to  1.  Hence  (Art.  287,  I) 
the  important  property  :  The  four  lines  «,/?,«+  7c/?, 
a  —  7c/9  form  an  harmonic  pencil. 

Also,  by  the  analogy  of  tangentials:  The  four  points 
«,  /9,  a  +  7c/9,  a  —  7c/9  are  harmonic. 

.  Theorem.  —  The  anharmonic  of  any  four  lines 
9,  a  +  Z/3,  a  -f-  m/9,  a  -f  nfi  is  equal  to 
(n  —  k)  (m—l) 


Let  OJ.,  OB  represent  the  lines 
a  and  ft  and  07f,  OJ^,  OM9  ON  the 
four  lines  a-f  ^A  a-Hft  a+wift  a+nft. 
Then,  if  rs  be  any  transversal,  the 
anharmonic  of  the  four  given  lines 
will  be  au.eo  :  ae.ou;  or,  what 
amounts  to  the  same  thing,  it  will  be 
An.  Ge.  25. 


256  ANALYTIC  GEOMETET. 

(ru — ra)  (ro  —  re)  :  (re — ra)  (ru  —  ro) .  Now,  taking  the 
lengths  of  the  perpendiculars  from  a,  e,  o,  u  upon  a,  both 
trigonometrically  and  from  the  equations  to  the  given 
lines,  we  obtain 

ru  =  —  nf)  cosec  Ore,     ra  =  - —  left  cosec  Ors, 
ro  =  — m-ft  cosec  Ors,     re  —  —  Z/3  cosec  Ors. 

Substituting  these  values  in  the  ratio  laslr  written,  and 
recollecting  that,  since  the  anharmonic  is  independent 
of  the  position  of  the  transversal,  we  may  take  rs  parallel 
to  the  line  /?,  and  thus  render  the  perpendicular  /9  a 
constant,  we  find  after  reductions 

R^  (n  —  k)  (m  —  I) 
(n  —  m)  (I  —  k) 

Remark. — The  student  can  easily  convince  himself 
that  the  harmonic  and  anharmonic  properties  above 
obtained  are  true  for  lines  whose  equations  are  of  the 
more  general  form  .L  =  0,  M=  0,  L -{-  kM=  0,  etc. 

2OO.  Theorem. — If  there  be  two  systems  of  right  lines, 
each  radiating  from  a  fixed  point,  and  if  the  several  mem- 
bers of  the  one  be  similarly  situated  with  those  of  the  other 
in  regard  to  any  two  lines  of  the  respective  groups,  then 
ivill  the  anharmonic  of  any  pencil  in  the  first  system  be 
equal  to  that  of  the  pencil  formed  by  the  four  correspond- 
ing lines  of  the  second. 

For,  in  such  a  case,  the  two  reference-lines  of  the 
first  group  being  L  =  0,  L'  =  0,  and  those  of  the  second 
M=Q,  Mf=§,  the  corresponding  pencils  will  be  (Z/+&Z/, 
L  -f  W,  L  +  mL',  L  +  nL')  and  (M  +  kM',  M+  IM', 
M-\-  mM ',  M  +  nM').  Hence,  the  theorem  follows 
directly  from  the  result  of  Art.  289. 


HOMOGRAPHIC  LINES.  257 

291.  Definition. — Systems  of  lines  whose  correspond- 
ing pencils  have  equal  anharmonics  are  called  homographic 
systems. 

Thus,  in  the  diagram  of  Art.  289,  the  two  systems 
(OA,  OK,  OL,  OM,  ON,  OB)  and  (0>A,  O'K',  O'L', 
O'M',  O'N',  O'B)  are  intended  to  represent  a  particular 
case  of  homographics. 

292.  In  the  examples  which  follow,  some   are   best 
adapted  for  solution  by  the  old  notation,  and  others  by 
the   abridged.     We  have  room   for   only  a  few   of  the 
manifold  properties  of  the  Right  Line. 

EXAM  PL  ES. 

1.  If  from  the  vertices  of  any  triangle  any  three  convergents  be 
drawn,  and  the  points  in  which  these  meet  the  opposite  sides  be 
joined  two  and  two  by  three  right  lines,  the  points  in  which  the 
latter  cut  the  sides  again  will  lie  on  one  right  line. 

2.  Any  side  of  a  triangle  is  divided  harmonically  by  one  of  the 
three  convergents  mentioned  in  Ex.  1,  and  the  line  joining  the  feet 
of  the  other  two. 

3.  In  the  figure  drawn  for  the  two  preceding  examples,  deter- 
mine by  tangentials  all  the  points  that  are  harmonic. 

4.  In  any  triangle,  the  two  sides,  the  line  drawn  from  the  vertex 
to  the  middle  of  the  base,  and  the  parallel  to  the  base  through  the 
vertex,  form  an  harmonic  pencil. 

5.  The  intersection  of  the  three  perpendiculars  to  the  sides  of  a 
triangle,  the  intersection  of  the  three  lines  drawn  from  the  vertices 
to  the  middle  points  of  the  sides,  and  the  center  of  the  circum- 
scribed circle,  lie  on  one  right  line. 

6.  APE,  CQD  are  two  parallels,  and  AP-.PB::  DQ:  QC:    to 
prove  that  the  three  right  lines  AD,  PQ,  BC  are  convergent. 

7.  From  three  points  A,  B,  D,  in  a  right  line  A  BCD,  three  con- 
vergents are  drawn  to  a  point  P;  and  through  C  is  drawn  a  right 
line  parallel  to  AP,  meeting  PB  in  E  and  PD  in  F:  to  prove  that 

AD.BC  :  AB. CD  ::  EC:  CF. 


258  ANALYTIC  GEOMETRY. 

8.  The  six  bisectors  of  the  angles  of  any  triangle  intersect  in 
only  four  points  besides  the  vertices. 

9.  If  through  the  vertices  of  any  triangle  lines  be  drawn  parallel 
to  the  opposite  sides,  the  right  lines  which  join  their  intersections 
to  the  three  given  vertices  will  meet  in  one  point.    [Use  both  notations 
in  succession.] 

10.  If  through  the  vertices  of  any  triangle  there  be  drawn  any 
three  convergents  whatever,  to  prove  that  these  three  lines  and  the 
three  sides  of  the  triangle  may  be  respectively  represented  by  the 

equations 

v  —  w  =  0,     w  —  u  =  Q,     u  —  v  =  0, 

v-\-  w  =  ?i,     w  -\-u  =  %,     u-\-v  =  %. 

11.  \tOAA' A",    OBE'R"  are   two    right    lines    harmonically 
divided,  the  former  in  A  and  A',  the  latter  in  .B  and  B',  the  lines 
AB,  A'B',  A//B//  either  meet  in  one  point  or  are  parallel. 

12.  If  on  the  three  sides  of  a  triangle,  taken  in  turn  as  diagonals, 
there  be  constructed  parallelograms  whose  sides  are  parallel  to  two 
fixed  right  lines,  the  three  remaining  diagonals  of  the  parallelograms 
will  meet  in  one  point. 

13.  The  three  external  bisectors  of  the  angles  of  any  triangle 
meet  the  opposite  sides  in  three  points  which  lie  on  one  right  line. 

14.  If  three  right  lines  drawn  from  the  vertices  of  any  triangle 
meet  in  one  point,  their  respective   parallels  drawn  through  the 
middle  of  the  opposite  sides  also  meet  in  one  point. 

15.  In  every  quadrilateral,  the  three  lines  which  join  the  middle 
points  of  the  opposite  sides  and  the  middle  points  of  the  diagonals, 
meet  in  one  point. 

16.  If  the  four  inner  angles  A,B,E,F  of  a  complete  quadrilateral 
(see  diagram,  Art.  219)  are  bisected  by  four  right  lines,  the  diago- 
nals of  the  quadrilateral  formed  by  these  bisectors  will  pass  through 
the  two  outer  vertices  of  the  complete  one,  namely,  one  through  C 
and  the  other  through  D. 

17.  Let  the  two  inner  diagonals  of  any  complete  quadrilateral 
(same  diagram)  intersect  in  O :  the  diagonals  of  the  two  quadri- 
laterals into  which  either  CO  or  DO  divides  ABEF,  intersect  in 
two  points  which  lie  on  one  right  line  with  O. 

18.  In  any  complete  quadrilateral,  any  two  opposite  sides  form 
an  harmonic  pencil  with  the  outer  diagonal  and  the  line  joining 
their  intersection  to  that  of  the  two  inner  diagonals. 


PROPERTIES  OF  THE  CIRCLE. 


259 


19.  Also,  the  two  inner  diagonals  are  harmonically  conjugate  to 
the  two  lines  which  join  their  intersection  to  the  two  outer  vertices. 

20.  Also,  two  adjacent  sides  are  harmonically  conjugate  to  their 
inner  diagonal  and  the  line  joining  their  intersection  to  that  of  the 
outer  diagonal  and  the  remaining  inner  one;  etc. 


CHAPTER   SECOND. 

THE   CIRCLE. 

203.  Before  attempting  the  discussion  of  the  three 
Conies  strictly  so  called,  it  will  be  advantageous  to 
illustrate  the  analytic  method  by  applying  it  to  that 
case  of  the  Ellipse  with  whose  properties  the  reader 
is  already  familiar  from  his  studies  in  pure  geometry : 
we  mean,  of  course,  the  Circle.  As  we  proceed  in  this 
application,  we  shall  be  enabled  to  define  those  elements 
of  curves  in  general,  which  constitute  at  once  the  leading 
objects  and  principal  aids  of  geometric  analysis. 

THE    AXIS    OF   X. 

294.  The  rectangular  equation  to  the  Circle  referred 
to  its  center  (Art.  136)  is 

a;2  +  y2  =  r2. 

If  we  solve  this  for  ?/,  we  obtain 
y  =  V(r+x)  (r — x).  But,  from 
the  diagram,  r  -\-  x  =  AM,  and 
r  —  x  =  MB ;  and  we  have  the 
well-known  property  of  Geom., 
325, 

The  ordinate  to  any  diameter  of  a  circle  is  a  mean 
proportional  between  the  corresponding  segments. 


260  ANALYTIC   GEOMETRY. 

295.  If  we  eliminate  between  the  equation  to  a  circle 
x2  -f  y2  =  r2  and  that  of  any  right  line  y  =  mx  -\-  b  by 
substituting    for  y   in    the   former   from   the    latter,   we 
shall  obtain,  as  determining  the  abscissas  of  intersection 
between  a  right  line  and  a  circle,  the  quadratic 

(1  -f-  m2)  x2  +  2mb. x  +  b2  —  r2  =  0. 

Now  the  roots  of  this  quadratic  are  real  and  unequal, 
equal,  or  imaginary,  according  as  (1  +  m2)  r2  is  greater 
than,  equal  to,  or  less  than  &2.  Hence,  when  the  first  of 
these  conditions  occurs,  the  right  line  will  meet  the  circle 
in  two  real  and  different  points ;  when  the  second,  in  two 
coincident  points  ;  when  the  third,  in  two  imaginary  points. 
Adhering,  then,  to  the  distinction  between  these  three 
classes  of  points,  we  may  assert,  with  full  generality, 

Every  right  line  meets  a  circle  in  two  points,  real,  coin- 
cident, or  imaginary. 

296.  It   is    so   important   that   the    distinction   just 
alluded    to   shall  be  exactly  understood  in   our   future 
investigations,    that    we    consider    it    worth    while    to 
illustrate  it  somewhat  more  at  length. 

I.  The  conception  of  two  real  points,  situated  at  a 
finite  distance  from  each  other,  is  of  course  already  clear 
to  the  student.  We  therefore  merely  add,  that  such  points 
are  sometimes  called  discrete,  or  discontinuous  points. 

II.  The  conception  of  coincident,  or,  as  they  are  more 
significantly  called,  consecutive  points,  is  peculiar  to  the 
analytic  method.  The  most  general  definition  of  con- 
secutive points  is,  that  they  are  points  whose  distance  from 
each  other  is  infinitely  small.  It  may  aid  in  rendering  this 
definition  clear,  to  think  of  two  points  which  are  drawing 
closer  and  closer  together,  which  tend  to  meet  but  not 


CONSECUTIVE  POINTS.  261 

to  pass  each  other,  and  whose  mutual  approach  is  never 
for  an  instant  interrupted.  The  distance  between  two 
such  points  is  evidently  less  than  any  assignable  quantity ; 
for  however  small  a  distance  we  may  assign  as  the  true 
one,  the  points  will  have  drawn  nearer  together  in  the 
very  instant  in  which  we  assign  it :  so  that  their  distance 
eludes  all  attempts  at  finite  statement,  and  can  only  be 
represented  by  the  phrase  infinitely  small. 

The  geometric  meaning  of  this  analytic  conception 
varies  with  its  different  applications.  Thus,  it  may  sig- 
nify exactly  the  same  thing  as  the  single  point  which,  in 
the  language  of  pure  geometry,  is  common  to  a  curve 
and  its  tangent.  For  since  the  distance  between  con- 
secutive points  is  infinitely  small ;  that  is,  so  small  that 
we  can  not  assign  a  value  too  small  for  it;  we  may 
assign  the  value  0,  and  take  the  points  as  absolutely 
coincident.  It  is  in  this  aspect,  mainly,  that  we  shall 
use  the  conception  in  our  future  inquiries.  Hence,  as 
from  the  infinite  series  of  continuous  values  which  the 
distance  between  two  consecutive  points  must  have,  we 
thus  select  the  one  corresponding  to  the  moment  of 
coincidence,  we  have  preferred  to  designate  the  concep- 
tion by  the  equivalent  and  for  us  more  pertinent  phrase 
coincident  points.  v 

The  student  should  be  sure  that  he  always  thinks  of 
the  distance  between  coincident  points  as  a  true  infinit- 
esimal. The  error  into  which  the  beginner  almost  always 
falls  is,  to  think  of  a  very  small,  instead  of  an  infinitely 
small,  distance.  He  thus  confounds  with  two  consecutive 
points,  two  discrete  ones  extremely  close  together,  between 
which  there  is  of  course  a  finite  distance.  The  conse- 
quence is,  that  he  finds  in  such  a  distance,  however  small, 
an  infinite  number  of  points  lying  between  his  supposed 
consecutives,  and  fancies  that  all  the  arguments  based 


262  ANALYTIC   GEOMETRY. 

on  the  conception  of  consecutiveness  are  fallacious. 
Whereas,  if  he  excludes  from  his  thoughts,  as  he  should, 
all  points  separated  by  any  finite  distance  however  small, 
he  will  have  points  absolutely  consecutive,  in  the  only 
sense  that  mature  reflection  attaches  to  that  term. 

III.  The  phrase  imaginary  points  is  also  peculiar  to 
analytic  investigation.  It  really  means,  wrhen  translated 
into  the  language  of  pure  geometry,  that  the  correspond- 
ing points  not  only  do  not  exist,  but  are  impossible.  But,  as 
we  have  mentioned  once  before,  the  expression,  together 
with  the  accessories  which  serve  to  carry  out  its  use,  is 
found  to  be  of  real  value  in  developing  certain  remote 
analogies  in  the  properties  of  curves.  We  shall  therefore 
retain  it,  only  cautioning  the  student  not  to  be  misled 
by  a  false  interpretation  of  it. 

297.  Definition. — A  Chord  of  any  curve  is  any  right 
line  that  meets  it  in  two  points. 

298.  Equation  to  a  Circular  Chord.— Let   the 

equation  to  the  given  circle  be  x2  -j-  y2  =  r2.  Since 
the  chord  passes  through  two  points  of  the  curve,  its 
equation  (Art.  95)  will  be  of  the  form 

y  —  y'  __  y"— y' . 

x  —  x'~~  x"—x'' 

in  which  xryr,  x"yrf,  since  they  both  lie  upon  the  circle, 
are  subject  to  the  condition 

2/2  -f  yn  =  r2  =  Xff2  _j_  ymf 

Hence,  xr*  —  xm  =  y"2  —  y'2\  and  we  obtain 

y"—yf  _ 
' 


CHORD  AND  DIAMETER.  263 

Therefore  the  required  equation  to  a  chord  is 

y  —  y'          xf  +  x"  . 

x  —  x'  =     ~  y'  +  y"  ' 

in  which  x'y',  x"y"  are  the  points  in  which  the  chord 
cuts  the  circle. 

Corollary, — By  a  course  of  analysis  exactly  similar  to 
that  just  used,  we  find  the  equation  to  any  chord  of  the 
circle  (x  — g)2  -f-  (y  — f)2  =  r2,  namely, 

y  —  y'  _        xf  +  x"  —  2g 
x  —  x'~    ~  y'  -f  y"  —  2/ 

in  which  x'y',  x"y"  are  the  intersections  of  the  circle 
with  its  chord,  and  gf  is  its  center. 

DIAMETERS. 

2OO.  Definition, — A  Diameter  of  any  curve  is  the 
locus  of  the  middle  points  of  parallel  chords. 

3OO.  Equation    to    a   Circular    Diameter. — To 

find  this,  we  must  form  the  equation  to  the  locus  of 
the  middle  points  of  parallel  chords  in  a  circle.  Let 
x9  y  be  the  co-ordinates  of  any  middle  point :  the 
formula  for  the  length  of  the  chord  from  xy  to  the 
point  x'y'  of  the  curve  (Art.  102)  gives  us  either 

x'  =x-\-  cl  or  y1  =  y  -f-  si. 

But  since  x'y'  is  a  point  on  the  circle  x2  -\-  y2  =  r2,  we 
have 

0  +  c/)2  +  (?/  +  s02  =  r2; 

or,  remembering  that  s2-f  c2  =  1>  we  get,  for  determining 
the  distance  I  of  the  point  xy  from  the  circle,  the  quadratic 

I2  +  2  (c-x  +  sy)  I  +  (x2  +  f  —  r2)  -0. 


264  ANALYTIC  GEOMETRY. 

Now,  as  xy  is  the  middle  point  of  a  chord,  the  two  values 
of  I  in  this  quadratic  must  be  numerically  equal  with 
contrary  signs.  Hence,  (Alg.,  234,  Prop.  3d,)  the  co- 
efficient of  I  must  vanish,  and  we  get 

ex  -f  sy  =  0. 

But,  in  the  present  inquiry,  s  and  c  are  the  sine  and 
cosine  of  the  angle  which  a  chord  through  xy  makes 
with  the  axis  of  x\  and  as  this  angle  is  the  same  for 
all  parallel  chords,  the  equation 

ex  -f-  sy  =  0 

is  a  constant  relation  between  the  co-ordinates  of  the 
middle  points  of  a  series  of  parallel  chords.  That  is, 
it  is  the  required  equation  to  any  diameter  of  the 
circle. 

301.  If  0  be  the  inclination  of  a  series  of  parallel 
chords  to  the  axis  of  x,  the  equation  just  obtained  may 
be  written  y  =  —  xcott).     Hence,  (Arts.  63;  96,  Cor.  3,) 
we  have  the  familiar  property,  4  <  ,.* 

Every  diameter  of  a  circle  passes  through  the  center, 
and  is  perpendicular  to  the  chords  which  it  bisects. 

Corollary. — From  this  we  immediately  obtain  the  im- 
portant principle :  If  a  diameter  bisects  chords  parallel  to 
a  second,  the  second  bisects  those  parallel  to  the  first. 

302.  Definition. — By  Conjugate  Diameters   of  a 

curve,  we  mean  two  diameters  so  related  that  each  bisects 
chords  parallel  to  the  other. 

303.  We  have  seen  (Art.  301)  that  the  equation  to 
any  diameter  of  a  circle  may  be  written  y  =  —  x  cot  0, 
in   which  expression  6  is  the  inclination  of  the  chords 


CONJUGATE  DIAMETERS.     TANGENT. 


265 


which   the   diameter  bisects.     Hence,  by  the  preceding 
definition,  the  equation  to  its  conjugate  will  be 


y  —  x  tan  6 


a). 


But,  putting  0'  ==  the  inclination  of  the  chords  bisected 
by  this  conjugate,  its  equation  will  take  the  form 


y  —  —  zcot  6' 


(2). 


Therefore,  as  the  condition  that  two  diameters  of  a  circle 
may  be  conjugate,  we  have,  since  (1)  and  (2)  are  only 
different  forms  of  the  same  equation, 


tan  6  tan  6'=— I 


(3): 


in  which  6  and  6'  may  be  taken  as  the  inclinations  of  the 
two  diameters  (since  these  are  each  perpendicular  to  the 
chords  which  they  bisect),  and  we  learn  (Art.  96,  Cor.  3) 
that 

The  conjugate  diameters  of  any  circle  are  all  at  right 
angles  to  each  other. 


THE    TANGENT. 

3O4.  Definition. — A  Tangent  of  any  curve  is  a  chord 
which  meets    it   in   two    coincident 
points. 

In  applying  this  definition,  the 
student  must  keep  in  mind  the 
principles  of  Art.  296,  II.  The 
annexed  diagram  will  aid  him  in 
apprehending  the  definition  cor- 
rectly. Let  PP"  be  any  chord 

passing   through   the    two    distinct   points  Pr    and   Pn ', 
and  let  PT  be  a  tangent  parallel   to   P'P".     Suppose 


266  ANALYTIC  GEOMETRY. 

P'P"  to  move  parallel  to  itself  until  it  coincides  with 
PT.  It  is  evident  that  as  P'P"  advances  toward  PT, 
the  points  Pf  and  P"  will  move  along  the  curve  toward 
each  other,  and  that  when  P'P"  at  length  coincides  with 
PT,  they  will  become  coincident  in  P,  which  is  called  the 
point  of  contact.  We  shall  often  allude  to  the  position 
of  the  two  coincident  points  through  which  a  tangent 
passes,  by  the  term  contact  alone. 

305.  Equation  to  a  Tangent  of  a  Circle. — Let 

the  given  circle  be  represented  by  x2  -J-  y2  =  r2.  Then, 
to  obtain  the  required  equation,  we  have  only  to  suppose, 
in  the  equation  to  any  chord  (Art.  298),  that  the  co-ordi- 
nates x'  and  x",  yr  and  y",  become  identical.  Making  this 
supposition,  reducing,  and  recollecting  that  x'2  -j-  y'2  =  r2, 
we  obtain 

x'x  +  y'y  =  r2 : 

in  which  xry'  is  the  point  of  contact. 

Corollary. — To  obtain  the  equation  to  a  tangent  of 
the  circle  (x — #)2  +  (y — f)2  =  r2-,  which  differs  from 
x*  -j-  y1  =  r2  only  in  having  the  origin  removed  to  the 
point  ( — </,  — /),  we  simply  transform  the  expression 
just  found  to  parallel  axes  passing  through  the  last- 
named  point,  by  replacing  x  and  y,  x'  and  y',  by  x  —  g, 
y—f,  x'  —  9i  y'—f-  We  thus  get 

(x'-g]  (x-g)  +  (y'-f)  (l/-f)=r2. 

We  may  also  obtain  this  less  directly,  by  applying  the 
condition  for  coincidence  to  the  equation  in  the  corollary 
to  Art.  298. 

306.  Condition  that  a  right  line  shall  touch  a 
Circle. — In  order  that  the  line  y  =  mx  -f-  b  may  touch 
the  circle  x2  -f-  y2  =  r2,  it  must  intersect  the  latter  in  two 


EQUATION  TO  TANGENT. 


267 


coincident  points ;  that  is,  the  co-ordinates  of  its  two  in- 
tersections with  the  circle  must  become  identical.  Hence, 
the  required  condition  will  be  found  by  eliminating  between 
y  —  mx  -\-  b  and  x2  -f  y2  =  r2,  and  forming  the  condition 
that  the  resulting  equation  may  have  equal  roots. 
The  resultant  of  this  elimination  is  the  quadratic 

(1  +  m2)  x2  -f  2mb. x  -f  (b2  —  r2)  =  0 ; 

and  (-see  third  equation  of  Art.  127,  et  seq.)  the  condition 
that  this  may  have  equal  roots  is 


Corollary. — Hence,  every  line  whose  equation  is  of  the 
form 

y  =  mx  J^r  j/1  -f-  w2, 

touches  the  circle  x2  -\-  y2==r2.  This  equation  belongs  to 
a  group  of  analogous  expressions  for  the  tangents  of  the 
several  Conies;  and,  on  account  of  its  great  usefulness, 
especially  in  problems  where  the  point  of  contact  is  not 
involved,  is  called  the  Magical  Equation  to  the  Tangent. 

3OT«  Auxiliary  Aiig-le. — In  problems  that  concern  the  in- 
tersection of  lines  with  a  circle,  it  is  often  advantageous  to  express 
the  co-ordinates  of  the  point  on  the  curve  in  terms  of  the  angle  which 
the  radius  drawn  to  such  point  makes  with  the  axis  of  a?.  By  so  doing, 
we  obtain  formulae  involving  only  one  variable. 

Thus,  if  V  =  the  angle  P'OM,  it  is  evi- 
dent that  we  shall  have,  for  the  co-ordi- 
nates of  any  point  P', 

x'  —  r  cos  tf,      y'=  r  sin  6/. 

In  this  notation,  the  equation  to  any 
chord  P'P"  becomes  (see  Art.  298) 

x  cos  J  (e"+V)  +  y  sin  $  (0"+?)  =rco3%  (V'— 


268  ANALYTIC  GEOMETRY. 

in  which  0"  —  the  angle  POM,  and  0"  =  the  angle  P"OM'. 
Hence,  the  corresponding  equation  to  any  tangent  is 

x  cos  0/  -f-  y  sin  &  =  r. 

This  may  also  be  obtained  by  substituting  directly  in  the  equation 
of  Art.  305;  and  it  expresses  (Art.  80,  Cor.)  the  well-known  property, 
that  the  tangent  to  a  circle  is  perpendicular  to  the  radius  at  contact. 

3O8.  To  draw  a  Tangent  to  a  Circle  from  a 
Fixed  Point.  —  This  problem,  so  far  as  it  belongs  to 
analytic  investigation,  requires  us  to  find  the  point  of 
contact  corresponding  to  the  tangent  which  passes 
through  the  arbitrary  point  x'y'. 

Let  x"y"  be  the  unknown  point  of  contact.  The 
equation  to  the  tangent  is  then  (Art.  305) 

x'rx  -f-  y"y  =  r2. 
But,  since  this  tangent  is  drawn  through  x'y',  we  have 

*v  +  y"y'  =  i*  (i). 

Moreover,  since  x"y"  is  upon  the  circle, 

=  r2  (2). 


Solving  for  x"  and  y"  between  (1)  and  (2),  we  get  the 
co-ordinates  of  the  point  of  contact,  namely, 


r,_r2x'±  ry'Vx'2+y'2—r2         ,  _  r2yf 


Corollary,  —  Hence,  through  any  fixed  point  there  can 
be  drawn  two  tangents  to  a  given  circle,  real,  coincident, 
or  imaginary.  Real,  when  x'2  -j-  y'2  >  r2  ;  that  is,  when 
x'yf  is  outside  the  circle.  Imaginary,  when  x'2  -j-  y'2  <  r2  ; 
that  is,  when  x'y'  is  within  the  circle.  Coincident,  when 
x12  +  y'2  =  r2  ;  that  is,  when  x'y'  is  on  the  circle. 


LENGTH  OF  TANGENT. 


269 


3O9.  Length  of  the  Tangent  from  any  point 
to  a  Circle.  —  Let  xy  be  any  point  in  the  plane  of  a 
circle  whose  center  is  the  point  gf.  Then  (Art.  51, 
I,  Cor.  1)  for  the  square  of  the  distance  between  xy 
and  gf,  we  have 


Putting  t  --  the  required  length  of  the  tangent,  we  get 
(since  the  tangent  is  perpendicular  to  the  radius  at 
contact)  t2  =  82  —  r2.  That  is, 

t2=(x-g)2+(y-f)2-r2. 

From  this  we  learn  that  if  the  co-ordinates  x  and  j  of 
any  point  be  substituted  in  the  equation  to  any  circle,  the 
result  will  be  the  square  of  the  length  of  the  tangent  drawn 
from  that  point  to  the  circle. 

Eemark.  —  We  have  shown  (Art.  142)  that  the  general 
equation 

A  (x2  +  if) 


is  equivalent  to  (x  —  g)2  +  (y  —  /)2  =  r2,  provided  we 
take  out  A  as  a  common  factor.  Hence,  the  property 
just  proved  applies  to  every  equation  denoting  a  circle, 
provided  it  be  reduced  to  the  form  in  question  by  the 
proper  division.  If,  then,  we  write  S  as  a  convenient 
abbreviation  for  the  left  member  of  the  equation  to  a 
circle,  in  which  the  common  co-efficient  of  x2  and  y2  is 
unity,  we  get 

t2  =  S. 

Corollary  —  The  square  of  the  length  of  the  tangent 
from  the  origin  of  the  circle. 

A  (x2  +  y2)  +  2Gx  +  2%  +  #  =  0, 

is  equal  to  C:  A:  that  is,  to  the  quotient  of  the  absolute 
term  by  the  common  co-efficient  of  x2  and  y2. 


270  ANALYTIC   GEOMETRY 

310.  Definition. — The  Subtangent  of  a  curve,  with 
respect  to  any  axis  of  rr,  is  the  portion  of  that  axis  inter- 
cepted between  the  foot  *  of  the  tangent  and  that  of  the 
ordinate  of  contact. 

Thus,  if  OT  is  consid- 
ered the  axis  of  x,  MT  is 
the  subtangent  correspond- 
ing to  PT  of  the  inner 
curve,  and  to  PT  of  the 
outer. 

311.  Subtangent   of  the   Circle. — To  obtain   the 
length  of  this,  we  find  the  intercept  OT,  cut  off  from  the 
axis  of  x  by  the  tangent,  and  subtract  from  it  the  abscissa 
of  contact  OM.     The  equation  to  the  tangent  being 

x'x  +  y'y  =  r\ 

to  find  the  intercept  on  the  axis  of  x,  we  make  y  —  0,  and 
take  the  corresponding  value  of  x.  Thus, 

r2 

x=-,=OT. 

x' 

Hence,  for  the  subtangent  MT,  we  have 

r2  —  x'2 

subtan  = - —  • 

x' 

That  is,  Any  subtangent  of  a  circle  is  a  fourth  proportional 
to  the  abscissa  of  contact  and  the  two  segments  into  which 
the  ordinate  of  contact  divides  the  corresponding  diameter. 

THE    NORMAL. 

312.  Definition. — The   Normal   of  a   curve   is   the 
right   line  perpendicular  to  a  tangent  at  the  point  of 
contact. 


*  The  point  in  which  a  line  meets  the  axis  of  x  is  termed  the  foot  of  the 
line.  Similarly,  the  point  where  a  line  meets  any  other  is  sometimes 
called  its  foot. 


NORMAL  AND  SUBNORMAL.  271 

313.  Equation  to  a  Xormal  of  a  Circle.  —  Since 
the  normal  is  perpendicular  to  the  line  x'x-\-y'y  =  r2,  and 
passes  through  the  point  of  contact  x'y',  its  equation  (Art. 
103,  Cor.  1)  is 


or,  after  reduction, 

y'x  —  x'y  =  0. 

The  form  of  this  expression  (Art.  95,  Cor.  2)  shows 
that  every  normal  of  a  circle  passes  through  the  center  :  — 
a  property  which  we  might  have  gathered  at  once  from 
the  definition. 

314.  Definition.  —  The  portion  of  the  normal  included 
between  the  point  of  contact  and  the  axis  on  which  the  cor- 
responding subtangent  is  measured,  is  called  the  length  of 
the  normal.    For  example,  Pf  0  in  the  diagram  of  Art.  310. 

From  the  result  of  Art.  313,  it  follows  that  the  length  of 
the  normal  in  any  circle  is  constant,  and  equal  to  the  radius. 

315.  Definition.  —  The  Subnormal  of  a  curve,  with 
respect  to  any  axis  of  x,  is  the  portion  of  that  axis  inter- 
cepted between  the  foot  of  the  normal  and  that  of  the 
corresponding  ordinate  of  contact. 

Thus,  in  the  diagram  of  Art.  310,  OM  is  the  subnormal 
corresponding  to  the  point  P  . 
Hence,  for  the  Circle,  we  have 

subnor  =  x'. 

That  is,  Any  subnormal  of  a  circle  is  equal  to  the  corre- 
sponding abscissa  of  contact. 

SUPPLEMENTAL    CHORDS. 

316.  Definition.  —  By  Supplemental  Chords  of  a 

circle,  we  mean  two  chords  passing  respectively  through 
the  extremities  of  a  diameter,  and  intersecting  on  the  curve. 
An.  Ge.  26. 


272  ANALYTIC  GEOMETRY. 

Thus,  AP,  BP  are  supplemental 
chords  of  the  circle  whose  center  is  0 
and  whose  radius  is  OA. 

317.   Condition  that  Chords  of  a 
Circle  be  Supplemental. — Take  for 
the  axis  of  x  the  diameter  through  whose 
extremities   the   chords   pass,  and  for  the  axis   of  y  a 
second   diameter   perpendicular  to   the  first.     Let  tp  - 
the  inclination  of  one  chord,  say  of  AP,  and  <p'  =  that 
of  the  other,  say  of  BP.     Then,  as  the  two  chords  pass 
through  the  opposite  extremities  of  the  same  diameter, 
their  equations  (Art.  101,  Cor.  1)  will  be 

y  =  (x  —  r)  tan  ^,     y—(x-\-  r)  tan  y>f. 

Hence,  at  their  point  of  intersection  (Art.  62,  Rem.)  we 
shall  have  the  condition 

yz  —  (x1  —  ?-2)  tan  (p  tan  cpf ; 

or,  since  they  intersect  upon  the  circle  xz-\-y2=r2, 
tan  <p  tan  <p'  =  —  1. 

Corollary. — From  the  form  of  this  condition  (Art.  96, 
Cor.  3)  we  infer  the  property : ,  Any  two  supplemental  chords 
of  a  circle  are  at  right  angles  to  each  other. 

This  is  only  another  way  of  stating  the  familiar  prin- 
ciple, that  every  angle  inscribed  in  a  semicircle  is  a  right 
angle. 

POLE    AND    POLAR. 

818.  The  terms  pole  and  polar  are  used,  as  already 
mentioned  (p.  238),  to  call  up  a  very  remarkable  relation 
between  points  and  right  lines,  which  depends  upon  a 
property  common  to  the  whole  order  of  Conies.  We 


POLE  AND  POLAR.  273 

shall  now  endeavor  to  develop  the  conception  of  the 
pole  and  polar  with  respect  to  the  Circle,  in  the  order 
according  to  which  they  naturally  appear  in  analysis. 

319.  Chord  of  Contact  belonging  to  Two  Tan- 
gents  which  pass  through   a   Fixed  Point. — By 

the  chord  of  contact  here  mentioned,  we  mean  the  right 
line  joining  the  two  points  of  contact  corresponding  to 
the  pair  of  tangents  which  (Art.  308,  Cor.)  we  have  seen 
can  be  drawn  to  a  circle  from  any  external  point.  Let 
x'y'  be  the  fixed  external  point,  and  x^y^  x2y2  the  two 
corresponding  points  of  contact.  The  equations  to  the 
two  tangents  (Art.  305)  will  then  be 

x\x + y\y  =  r*i   xix + y$ = rZ- 

Now,  since  both  the  tangents  pass  through  x'yr,  we  have 
the  two  conditions 

x,x'  +  y,y'  =  r\     x2x'  -f  y2y'  =  r2. 

Hence,  the  co-ordinates  of  both  points  of  contact  satisfy 
the  equation 

x'x  -f  y'y  =  r2 : 

which  is  therefore  the  equation  to  the  chord  of  contact. 

320.  Locus  of  the  intersection  of  Tangents  at 
the   extremities   of  a   Chord   passing   through  a 
Fixed   Point. — Let    x'y'   be  the   fixed   point   through 
which  the  chord  passes,  and  x}y\  the  point  in  which  the 
two  tangents   drawn  at  its   extremities  intersect.     The 
equation  to  the  chord  (Art.  319)  will  be  x}x  Jry\y  =  r2", 
and,  as  x'y'  is  a  point  on  the  chord,  we  shall  have  the 
condition 

*\d  +  y\y'  =  r2, 


274  ANALYTIC   GEOMETRY. 

no  matter  what  be  the  direction  of  the  chord.  Hence, 
supposing  the  chord  to  be  movable,  and  the  intersection 
of  the  two  corresponding  tangents  to  be  a  variable  point, 
the  co-ordinates  of  the  latter  will  always  satisfy  the 
equation 

x'x  +  y'y  =  r : 

which  is  therefore  the  equation  to  the  locus  required, 
and  shows  (Art.  85)  that  this  locus  is  a  right  line. 

321.  Relation  of  the  Tangent  to  this  Locus 
and  to  the  Chord  of  Contact. — The  equation  to  the 
chord  of  contact  of  the  two  tangents  drawn  from  any 
point  outside  of  a  circle,  is 

x'x  +  y'y  =  r>  (1), 

in  which  x'y'  is  the  point  from  wliicli  the  tangents  are 
drawn.  The  equation  to  the  locus  of  the  intersection 
of  two  tangents  drawn  at  the  extremities  of  any  chord 
passing  through  a  fixed  point,  is 

x'x  +  yfy=r*  (2), 

in  which  x'y'  is  the  point  through  which  the  movable  chord 
is  drawn.  The  equation  to  any  tangent  of  the  same  circle 
to  which  the  chord  (1)  and  the  locus  (2)  refer,  is 

x'x  +  y'y  =  r*  (3), 

in  which  x'y1  is  the  point  of  contact.  What,  then,  is  the 
significance  of  this  remarkable  identity  in  the  equations 
to  these  three  lines  ?  It  certainly  means  that  there  is 
some  law  of  form  common  to  the  tangent,  the  chord  of 
contact,  and  the  locus  mentioned.  For  (1),  (2),  (3)  assert 
that  the  chord  of  contact  is  connected  with  the  point 
from  which  the  two  corresponding  tangents  are  drawn, 
and  that  the  right  line  forming  the  locus  of  the  inter- 


POLE  AND  POLAR.  275 

section  of  two  tangents  at  the  extremities  of  a  chord 
passing  through  a  fixed  point  is  connected  with  that 
point,  in  exactly  the  same  way  that  the  tangent  is  con- 
nected with  its  point  of  contact.  Now  a  right  line  in 
the  plane  of  a  circle  must  be  either  a  tangent,  a  chord 
of  contact,  or  a  locus  corresponding  to  (2)  :  hence  we 
learn  that  the  Circle  possesses  the  remarkable  property 
of  imparting  to  any  right  line  in  its  plane  the  power 
of  determining  a  point  ;  and  reciprocally. 

This  property  is  known  as  the  principle  of  polar  re- 
ciprocity ;  or,  as  it  is  sometimes  called,  the  principle  of 
reciprocal  polarity.  It  is  fully  expressed  in  the  following 
twofold  theorem  : 

I.  If  from  a  fixed  point  chords  be  drawn  to  any  circle, 
and  tangents  to  the  curve  be  formed  at  the  extremities  of 
each  chord,  the  intersections  of  the  several  pairs  of  tangents 
will  lie  on  one  right  line. 

II,  If  from  different  points  lying  on  one  right  line  pairs 
of  tangents  be  drawn  to  any  circle,  their  several  chords  of 
contact  will  meet  in  one  point. 

The  truth  of  (I)  is  evident  from  the  equation  of  Art.  320  ; 
that  of  (II)  appears  as  follows  :  —  Let  Ax-{-By-\-C=Q  be 
any  right  line.  The  chord  of  contact  corresponding  to 
any  point  x'y'  of  this  line  (Art.  319)  is  xfx-\-  yfy  =  r2. 
Now  the  co-efficients  of  this  equation  are  connected  by 
the  linear  relation 

Ax' 


and  the  chord  passes  through  a  fixed  point  by  Art.  117. 
We  shall  find,  as  we  .  go  on,  that  the  property  just 
proved  of  the  Circle  is  common  to  all  conies.  The 
reciprocal  relation  between  a  point  and  a  right  line  is 
expressed  by  calling  the  point  the  pole  of  the  line,  and 
the  line  the  polar  of  the  point. 


276 


ANALYTIC  GEOMETRY. 


322.  Definitions, — The  Polar  of  any  point,  with 
respect  to  a  circle,  is  the  right  line  which  forms  the  locus 
of  the  intersection  of  the  two  tangents  drawn  at  the  ex- 
tremities of  any  chord  passing  through  the  point.  Thus, 
LL'L"  is  the  polar  of  P. 

The  Pole  of  any  right  line,  with  respect  to  a  circle, 
is  the  point  in  which  all  the  chords  of  contact  corre- 
sponding to  different  points  on  the  line  intersect  each 
other.  Thus,  P  is  the  pole  of  LL'L". 

These     definitions 
enable    us    to    con- 
struct the  polar  when 
the    point   is    given, 
or  the  pole  when  we 
have  the  line.    Thus, 
if  P  be  the  point,  we 
draw  any  two  chords 
through  it,  as  MPN, 
OPQ,  and  the  corre- 
sponding pairs  of  tan- 
gents, ML,NL;OL", 
QL".  The  line  which 
joins  the  points  L  and 
L",  in  which  the  re- 
spective pairs  of  tangents  intersect,  is  the  polar  of  P. 
On  the  other  hand,  if  LL"  be  the  given  line,  we  draw 
from  any  two  of  its  points,  as  L  and  Lff,  two  pairs  of 
tangents,  LM,  LN;  L"0,  L"Q,  and  the  two  correspond- 
ing chords  of  contact,  MN,  OQ:  the  point  P  in  which 
the  latter  meet,  is  the  required  pole. 

We  have  given  the  construction  in  the  form  above  be- 
cause it  answers  in  all  cases.  It  is  evident,  however,  from 
the  results  of  Art.  321,  that  when  P  is  without  the  circle, 
its  polar  is  the  corresponding  chord  of  contact  QR;  and 


CONSTRUCTION  OF  POLAR.  277 

that  when  it  is  on  the  curve,  its  polar  is  the  corresponding 
tangent  TS.  In  these  cases,  then,  our  drawing  may  be 
modified  in  accordance  with  these  facts. 

323.  From  all  that  has  now  been  shown,  it  follows 
that  the  equation  to  the  polar  of  any  point  x'y',  with 
respect  to  the  circle  x2  -j-  y2  =  r2,  is 

x'x  -j-  y'y  =  r2. 

Now  the  equation  to  the  line  which  joins  x'y'  to  the  center 
of  the  same  circle  (Art.  95,  Cor.  2)  is 


Hence,  (Art.  99,)  The  polar  of  any  point  is  perpendicular 
to  the  line  which  joins  that  point  to  the  center  of  the  corre- 
sponding circle. 

Corollary  —  This  property  affords  a  method  of  con- 
structing the  polar,  simpler  than  that  explained  in  the 
preceding  article.  For  (Art.  92,  Cor.  2)  the  distance 
of  the  polar  from  the  center  of  its  circle  is 

r2 


in  which  (Art.  51,  I,  Cor.  2)  l/V2  -f  yr*  is  the  distance 
of  the  pole  from  the  center ;  hence,  To  construct  the  polar, 
join  the  pole  to  the  center  of  the  circle,  and  from  the  latter 
as  origin  lay  off  upon  the  resulting  line  a  distance  forming 
a  third  proportional  to  its  ivhole  length  and  the  radius: 
the  perpendicular  to  the  first  line,  drawn  through  the  point 
thus  reached,  ivill  be  the  polar  required. 

In  the  next  four  articles,  we  will  present  a  few  striking 
properties  of  polars. 


278  ANALYTIC  GEOMETRY. 


The  condition  that  a  point  a/y  shall  lie  upon  the  polar 
of  x'y'  is  of  course 


Now,  obviously,  this  is  also  the  condition  that  xfy/  shall  lie  upon 
the  polar  of  x"y" .  Therefore,  If  a  point  lie  upon  the  polar  of  a 
second,  the  second  will  lie  upon  the  polar  of  the  first. 

Corollary — Hence,  The  intersection  of  two  right  lines  is  the  pole 
of  the  line  which  joins  their  poles. 

Remark. — We  shall  find  hereafter  that  these  properties  are  com- 
mon to  all  conies. 

325.  The  distance  of  x'y'  from  the  polar  of  a/'/'  (Art.  105, 
Cor.  2)  is 

P'  =  : 

and  the  distance  of  a/'y"  from  the  polar  of  a/y  is 
sV'+vV'-r2 


Hence,  The  distances  of  two  points  from  each  other  s  polars  are  pro- 
portional to  their  distances  from  the  center  of  the  corresponding  circle. 

326.  Definitions — Two  triangles  so  situated  with  respect  to 
any  conic  that  the  sides  of  the  one  are  polars  to  the  vertices  of  the 
other,  are  called  conjugate  triangles. 

Thus,  in  the  diagram,  AE C  and 
abc  are  conjugate  triangles  with 
respect  to  the  circle  EQS. 

The  corresponding  sides  of  two 
conjugate  triangles  are  those  sides 
of  the  second  which  are  opposite  to 
the  poles  of  the  sides  of  the  first,  and  A<*^ 
reciprocally.  The  corresponding  an- 
gles lie  opposite  to  the  polars  of  the 

several  vertices.    Thus,  AE  and  ab  are  corresponding  sides;  A  and  a, 
corresponding  angles ;  etc. 

A  triangle  whose  sides  are  the  polars  of  its  own  vertices  is  called 
self -con jugate.  To  draw  a  self-conjugate  triangle,  take  any  point  P, 
and  form  its  polar;  on  the  latter,  take  any  point  T,  and  form  its 
polar:  this  new  polar  (Art.  324)  will  pass  through  P,  and  will  of 


CONJUGATE  TRIANGLES.  279 

course  intersect  the  polar  of  P  in  some  point  Z;  join  P27,  and 
PTZ  will  be  the  required  triangle.  For  TZ  is  the  polar  of  P,  and 
ZP  of  T,  by  construction;  while  PT  is  the  polar  of  Z  by  the  cor- 
ollar to  Art.  324. 


Theorem.  —  The  three  lines  which  join  the  corresponding 
vertices  of  two  conjugate  triangles  meet  in  one  point. 

Let  the  vertices  of  one  triangle  be  xfy',  x"y",  x///y///  ;  the  sides 
of  the  other  will  then  be 

x'x  +  y'y  =  r\      y/'x  +  y"y  =  r\      x'"x  -f  y'"y  =  r*. 

For  brevity,  write  these  equations  Px=  0,  Pxx=  0,  Pxxx^=  0  ;  and  let 
PIX/,  P/x/  denote  the  results  of  substituting  x'y*  in  Pxx  and  Pxxx  ; 
P/xx,  P/,  the  results  of  substituting  *"y"  in  P"'  and  Px;  and 
P3X,  P3X/,  the  results  of  substituting  y/"y"'  in  Px  and  P/x.  For  the 
three  lines  joining  the  corresponding  vertices  (see  diagram,  Art.  326) 
we  shall  then  have  (Art.  108,  Cor.  1) 

P/".P"  —  P/x  .P///=  0  (Aa), 

p,   .p't'—pv'.p'  =o 

P/X.PX  -P/  .PXX-O 


Now,  writing  the  abbreviations  in  full,  we  get  Pl///=P./;  P/x—  P/; 
P2XXX—  P/x.  Hence,  the  three  equations  just  written  vanish  iden- 
tically when  added,  and  the  proposition  is  proved. 

Corollary  —  By  Art.  281,  it  follows  that  the  intersections  of  the 
corresponding  sides  of  two  conjugate  triangles  lie  on  one  right  line. 
That  is,  conjugate  triangles  are  homologous. 


SYSTEMS   OF  CIRCLES. 
I.  SYSTEM  WITH  COMMON  RADICAL  AXIS. 

328.  Any  two  circles  lying  in  the  same  plane  give 
rise   to   a   very  remarkable  line,  which   is   called   their 
radical  axis.     Its  existence  and  its  fundamental  property 
will  appear  from  the  following  analysis: 
An.  Ge.  2T. 


280  ANALYTIC  GEOMETRY. 

Let  S=Q,  S'=Q  be  the  equations  to  two  circles,  so 
written  that  in  each  the  common  co-efficient  of  x2  and  y1 
is  unity.  Then  will  the  equation  S  +  kSr=Q  in  general 
denote  a  circle  passing  through  the  points  in  which  >S  and 
Sf  intersect;  for  in  it  x2  and  y1  will  have  the  common 
co-efficient  (1  -f-  7c),  and  obviously  it  will  vanish  when  S 
and  iSf  vanish  simultaneously.  To  this  theorem,  how- 
ever, there  is  one  exception,  namely,  when  Jc=  —  1. 
The  resulting  equation  is  then 


and,  being  necessarily  of  the  first  degree,  denotes  a  right 
line. 

Moreover,  this  equation  is  satisfied  by  an  infinite 
series  of  continuous  values,  whether  any  can  be  found 
to  satisfy  S  and  Sf  simultaneously  or  not.  That  is,  The 
line  S  —  S'  is  real  even  when  the  two  common  points  of  the 
circles,  through  ivhich  it  passes,  are  imaginary.  When  the 
circles  intersect  in  real  points,  the  line  is  of  course  their 
common  chord;  and  it  might  still  be  called  by  that  name 
even  when  the  points  of  intersection  are  imaginary,  if  we 
chose  to  extend  the  usage  in  regard  to  imaginary  points 
and  lines  which  has  been  so  frequently  employed.  To 
avoid  this  apparent  straining  of  language,  however,  the 
name  radical  axis  has  been  generally  adopted. 

The  equation  S=Sf  asserts  (Art.  309,  Rem.)  that  the 
tangents  to  S  and  /S",  drawn  from  any  point  in  its  locus, 
are  equal.  Hence,  The  radical  axis  of  two  circles  is  a 
right  line,  from  any  point  of  which  if  tangents  be  drawn 
to  both  of  them,  the  two  tangents  will  be  of  equal  length. 

329.  Writing  S  —  S'  in  full,  and  reducing,  the  equa- 
tion to  the  radical  axis  becomes  (Art.  134) 

'!+fn)  }• 


RADICAL  AXIS. 


281 


Now  the  equation  to  the  line  joining  the  centers  of  the 
two  circles  (Art.  95,  Cor.  1)  may  be  written 


Therefore,  (Art.  99,  Cor.,)  The  radical  axis  of  two  circles 
is  perpendicular  to  the  line  which  joins  their  centers* 

Corollary.  —  Hence,  To  construct  the  radical  axis  of  two 
circles  ,  find  its  intercept  on  the  axis  of  x  by  making  j  =  0 
in  the  equation  S  —  S'—  0,  and  through  the  extremity  of  the 
intercept  draw  a  perpendicular  to  the  line  of  the  centers. 

Remark.  —  This  construction  is  applicable  in  all  cases; 
but,  when  the  circles  intersect  in  real  points,  the  axis  is 
obtained  at  once  by  drawing  the  common  chord. 

33O.  If  S,  Sr,  S"  be  any  three  circles,  the  equations 
to  the  three  radical  axes  to  which  the  group  gives  rise 
will  be 


which  evidently  vanish  identically  when  added.  Hence, 
The  three  radical  axes  belonging  to  any  three  circles  meet 
in  one  point,  called  the  RADICAL  CENTER. 

Corollary.  —  We  may  therefore  construct  the  radical 
axis  as  follows  :  Find  the 
radical  center  of  the  two 
given  circles  witli  respect  to 
any  third,  and  through  it 
draw  a  perpendicular  to  the 
line  of  their  centers.  The 
annexed  diagram  will  illus- 
trate the  details  of  the  pro- 
cess. In  it,  c  and  c'  are  the 
centers  of  the  two  given  cir- 
cles, C  the  radical  center,  and  CQC'  the  radical  axis. 


282 


ANALYTIC   GEOMETRY. 


331.  Two  special   cases  of  the  radical  axis  deserve 
notice.     First:  We  have  seen  that  a  point  may  be  re- 
garded  as   an    infinitely  small    circle.     Hence,   a   point 
and   a   circle  have   a   radical   axis;    that  is,  given   any 
point  and  any  circle,  we  can  always  find  a  right  line, 
from  any  point  of  which  if  we  draw  a  tangent  to  the 
circle  and  a  line  to  the  given  point,  the  two  will  be  of 
equal   length.     The  axis  is  of  course   perpendicular   to 
the  line  drawn  from  the  given  point  to  the  center  of  the 
circle.     It  lies  without  the  circle,  whether  the  point  be 
within  or  without ;  for,  as  the  radical  axis  always  passes 
through  the  points  common  to  its  two  circles,  if  it  cut  the 
given  circle,  the  given  point  would  form  two  consecutive 
points  of  that  curve.     From  this  it  appears,  that,  when 
the  given  point  is  on  the  given  circle,  the  axis  is  the 
tangent  at  the  point. 

Second :  If  both  circles  to  which  a  radical  axis  belongs 
become  points,  we  have  a  line  every  point  of  which  is 
equally  distant  from  two  given  ones.  Hence,  the  radical 
axis  of  two  points  is  the  perpendicular  bisecting  the  dis- 
tance between  them. 

We  now  proceed  to  the  properties  of  the  entire  system 
of  circles  formed  about  a  common  radical  axis. 

332.  Definition. — By  a  System  of  Circles  with  a  Common 
Radical   Axis,   we 

mean  a  system  so  sit- 
uated with  respect  to 
a  fixed  right  line, 
that,  if  a  tangent  be 
drawn  to  each  circle 
from  any  point  in 
the  line,  all  these 
tangents  will  be  of 
equal  length. 

The  simplest  case  of  such  a  system  is  that  of  the  infinite  series 
of  circles  which  can  be  passed  through  two  given  points.  The 


CIRCLES  WITH  COMMON  AXIS.  283 

diagram  illustrates  this  case.  Another  is  that  of  a  series  of  circles 
touching  each  other  at  a  common  point.  The  appearance  of  the 
system  when  the  two  common  points  are  imaginary,  will  be  pre- 
sented farther  on,  in  connection  with  the  method  of  constructing 
the  system. 

It  follows  directly  from  Art.  329,  that  all  the  centers  of  the  system 
lie  on  one  rig  Jit  line  at  right  angles  to  their  radical  axis. 

333*  Equation  to  any  member  of  the  System.  —  The  equa- 
tion to  any  circle  whose  center  lies  on  the  axis  of  #,  at  a  distance  g 
from  the  origin,  may  be  written  (Art.  134) 

x*  +  f-2gx  =  r2  —  g1', 

so  that  the  circle  will  cut  the  axis  of  y  in  real  or  imaginary  points 
according  as  r2  —  g2  is  positive  or  negative,  the  quantity  Vr'1  —  g'1 
representing  half  the  intercept  on  the  axis  of  y.  Hence,  if  in  the 
system  of  circles  with  a  common  radical  axis,  the  common  line  of 
centers  be  taken  for  the  axis  of  x,  and  the  common  radical  axis  for 
the  axis  of  y  :  by  putting  k  =  the  arbitrary  distance  of  the  center 
from  the  origin,  and  o2—  constant  =  r2  —  &2,  we  may  write  the  equa- 
tion to  any  member  of  the  system 


and  the  corresponding  system  will  cut  the  radical  axis  in  real  or 
in  imaginary  points  according  as  o2  is  positive  or  negative. 

Corollary,  —  Hence,  To  trace  the  system  from  the  equation,  assume 
different  centers  corresponding  to  arbitrary  values  of  k,  and  from  them, 
with  radii  in  each  case  equal  to  y  k*  d=  ^2,  describe  circles. 

334.  The'  Orthogonal  Circle.  —  From  the  definition  of  the 
system  (Art.  332),  it  follows  that  the  locus  of  the  point  of  contact 
of  the  tangent  drawn  from  any  point  in  the  common  radical  axis 
to  any  member  of  the  system,  is  a  circle.  Since,  then,  the  corre- 
sponding tangents  of  the  system  are  all  radii  of  this  circle,  tangents 
to  this  circle  at  the  points  where  it  cuts  the  several  members  of  the 
system  will  be  perpendicular  to  the  respective  tangents  of  the  system. 
In  other  words,  the  circle  in  question  cuts  every  member  of  the 
system  at  right  angles,*  and  may  therefore  be  called  the  orthogonal 
circle  of  the  system. 


-;:  The  angle  between  two  curves  is  the  angle  contained  by  their  re- 
spective tangents  at  the  point  of  intersection. 


284 


ANALYTIC  GEOMETRY. 


Since  the  center  of  such  a  circle  is  any  point  on  the  radical  axis, 
there  is  an  infinite  series  of  orthogonal  circles  for  every  system  with 
a  common  axis.  But  for  the  special  purpose  to  which  we  are  about 
to  apply  it,  any  one  of  these  may  be  selected,  of  which  we  shall 
speak  as  the  orthogonal  circle. 

335.  Construction  of  tlic  System. — We  can  now  construct 
the  system  geometrically,  in  all  cases.  The  only  case  that  needs 
illustration,  however,  is  that  of  the  system-  passing  through  two 
imaginary  common  points.  Since  (Art.  334)  the  tangents  of  the 
orthogonal  circle  are  all  radii  of  the  circles  forming  the  system,  we 
may  draw  any  number  of  these  circles  as  follows :  — Lay  down  any 
right  line  MN,  and  any  perpendicular  to  it  HQ.  On  the  latter, 


take  any  point  C  as  a  center,  and,  with  any  radius  (7Q,  describe  a 
circle  cutting  MN  in  the  points  m  and  n.  At  any  points  a,  6,  c,  d,  e 
of  this  circle,  draw  tangents  to  meet  MN  in  1,  2,  3,  4,  5;  and  from 
these  points  as  centers,  with  radii  in  each  case  equal  to  the  corre- 
sponding tangent,  describe  circles.  It  is  evident  that  the  radii  Ca, 
(76,  Cfc,  etc.,  of  the  fundamental  circle,  w'ill  all  be  tangents  to  the 
respective  circles  last  drawn.  Hence  HQ,  is  the  common  radical 
axis  of  all  these  circles,  and  the  circle  C-Qmn  is  orthogonal  to 
them. 

33G.  Properties  of  the  System:  Limiting:  Points — From 
the  nature  of  the  foregoing  construction  and  the  resulting  diagram, 
we  obtain  the  following  properties : 


PONCELETS  LIMITING  POINTS.  285 

I.  The  circles  of  a  system  having  two  imaginary  common  points, 
in  which  the  orthogonal  circle  cuts  the  line  of  centers  in  two  real 
points  m  and  w,  exist  in  pairs :  to  every  circle  on  the  right  of  the 
radical  axis,  corresponds  an  equal  one  on  the  left,  with  an  equally 
distant  center. 

II.  In  a  system  of  this  character,  the  center  of  no  circle  can  lie 
nearer  to  the  radical  axis  than  m  or  n.     For  the  radius  of  the  vari- 
able member  of  the  system  continually  diminishes  as  the  tangent 
of  the  orthogonal  circle  advances  from  Q  toward  m  and  n,  and  at 
m  and  n  it  vanishes. 

III.  But  the  center  of  a  member  of  the  system  may  be  as  remote 
from  the  radical  axis  as  we  please.     For  the  tangents  of  the  orthog- 
onal circle  at  Q  meet  the  line  of  centers  at  infinity. 

IV.  Hence,  the  points  m  and  n,  and  the  axis  RQ  form  the  inferior 
and  superior  limits  of  the  system;  in  short,  are  the  corresponding 
limiting  members  of  it :  m  and  n  being  equal  infinitesimal  cirqles  at 
equal  distances  from  the  axis,  and  the  axis  itself  being  the  resultant 
of  two  coincident  circles  having  equal  infinite  radii. 

V.  The  circle  C-Qmn  is  drawn  in  the  diagram  to  cut  MN  in  real 
points ;  but  if  the  student  will  draw  a  new  diagram,  in  which  C-Qmn 
fails  to  cut  MN,  he  will  find  that  the  circles  of  the  resulting  system 
all  cut  each  other  in  two  real  points  on  the  line  RQ.     Hence,  a 
system  of  circles  with  a  common  radical  axis  intersect  each  other 
in  two  real  or  imaginary  points,  according  as  the  limiting  points  m 
and  n  are  imaginary  or  real. 

VI.  The  limiting  points  m  and  n  are  by  construction  equally 
distant  from  every  point  in  RQ.     Moreover,  every  orthogonal  circle 
is  described  from  some  point  in  RQ,  and  cuts  every  member  of  the 
system  at  right  angles.    Hence  every  orthogonal  circle  passes  through 
the  limiting  points.  .  That  is,  The  orthogonal*  of  any  system  of  circles 
with  a  common  radical  axis,  form  a  complemental  system,  whose  radical 
axis  is  the  line  joining  the  centers  of  the  conjugate  system. 

VII.  Hence,  if  a  system  of  circles  intersect  in  two  real  points, 
the  conjugate  system  of  orthogonal   circles   will  intersect  in   two 
imaginary  ones;  and  reciprocally. 

337«  The  Limiting  Points  by  Analysis. — If  a  system  of 
circles  cut  its  radical  axis  in  two  imaginary  points,  the  equation  to 
any  member  of  the  system  (Art.  333)  is 

z2  +  </2-2/vZ  =  -o'  (1), 


286  ANALYTIC  GEOMETRY. 

in  which  J2  is  constant  for  the  whole  system,  while  k  varies  for  each 
different  member.  Now  —  #2  =  r2  —  A2  .*.  r=  V  k~  —  ^  :  therefore 
?•  vanishes  when  k  =  ±  tf,  and  becomes  imaginary  when  7;  <  6  or 
>  —  6.  Hence,  the  two  points  (y  =  0,  x  =  6)  and  (y  =  0.  x  —  —  (5) 
are  the  infinitesimal  circles  which  we  have  called  the  limiting  points 
of  the  system;  for  we  have  just  shown  that  they  have  the  property 
of  Art.  336,  II,  and  they  are  represented  (Art.  61,  Hem.)  by  the 
equation 

(a;  =F  0)2  -f-  f  =  0  ;    that  is,    z2  +  f  =F2  dx  =  —  <P, 

which  conforms  to  the  type  of  (1). 

To  exhibit  the  singular  nature  of  these  limiting  points,  we  will 
now  develop  one  more  property  of  the  system  to  which  they  belong. 
The  equation  to  any  of  its  members  may  be  thrown  into  the  form 


Hence,  (Art.  305,  Cor.,*)  the  polar  of  any  point  sfy',  with  respect 
to  any  member  of  the  system,  will  be  represented  by 


tfx  +  y'y  +  F  —  k  (x  -f  *0  =  0  (2). 

Now  (Art.  108)  the  line  denoted  by  (2)  passes  through  the  inter- 
section of  the  two  lines  x'x  -f  y'y  +  &  =  0  and  x  -f  x*=  0,  whatever 
be  the  value  of  k.  Therefore,  If  the  polar  s  of  a  given  point  be  taken 
with  respect  to  the  whole  system  of  circles  having  a  common  radical  axis, 
they  will  all  meet  in  one  point. 

Suppose,  then,  that  x'tf  be  either  of  the  limiting  points.     The 
polar  will  then  become 

x  -  ^F  6  (3). 

Hence,  The  polar  of  either  limiting  point  is  a  line  drawn  through  the 
other  at  riyht  angles  to  the  line  of  centers,  and  is  therefore  absolutely 
fixed  for  the  whole  system. 

II.    TWO  CIRCLES  WITH  A  COMMON  TANGENT. 

338.  The  problem  of  constructing  a  common  tangent 
to  two  given  circles  (which  properly  belongs  to  Deter- 


*  The  student  will  remember  that  the  equations  to  the  tangent  and  polar 
of  the  Circle  are  identical  in  form. 


CIRCLES  WITH  COMMON  TANGENTS.  287 

minate  Geometry,  and  which  we  solved  under  that  head 
on  pp.  18  —  21)  leads  to  some  important  results  when 
treated  by  the  methods  of  Indeterminate  Geometry.  A 
few  of  these,  we  shall  now  present. 

339.  The  problem,  as  coming  within  the  sphere  of  pure 
analysis,  consists  in  finding  co-ordinates  of  contact  such  that  the 
corresponding  tangent  may  touch  both  circles.  Suppose,  then,  that 
the  equations  to  the  two  circles  are 


The  equation  to  a  tangent  of  S  will  then  be  (Art.  305,  Cor.) 
(x'-g)  (x-g}  +  (;/-/)  (y-/)-r2; 

and  our  problem  is,  so  to  determine  x'y'  that  this  line  may  also 
touch  S/. 

In  settling  what  condition  x'y'  must  satisfy  in  order  that  this 
result  may  take  place,  it  will  be  convenient  to  employ  the  auxiliary 
angle  mentioned  in  Art.  307.  Let  6  =  the  inclination  of  the  radius 
through  x'y'  \  then  will  x'  —  #  =  7-0080,  and  y'  —  f—rsin6.  The 
equation  to  the  tangent  of  S  may  therefore  be  written 


sin0-f-r  (1), 

and,  similarly,  the  equation  to  a  tangent  of  $', 

x  cos  0"  +  y  sin  ^  =/  cos  0"  +/'  sin  0/  +  r'  (2). 

Now  (1)  will  represent  the  same  line  as  (2),  if  the  mutual  ratios 
of  its  co-efficients  are  the  same  as  those  of  (2)  ;  that  is,  if  simul- 
taneously 

tan  6  =  tan  0^ 

(g  cos  0  +/sin  6  -f  r)  cos  6/  ==  (/  cos  0/  +/sin  0/  -f  r')  cos  0. 

The  first  of  these  conditions  is  satisfied  either  by  O'=0,  or  (P=ir-\-6. 
Combining  the  two,  then,  on  both  suppositions,  we  obtain 

(  g>—g)  cos  6  +  (/'—/)  sin  6  =  r  —  r', 
(/-#)  cos  0  +(/'-/)  sin  0  =  r  +  r'  ; 


288 


ANALYTIC  GEOMETRY. 


x' q  ?// f 

or,  after  replacing  cos  0  and  sin  0  by  their  values, and  - — — 

(/-</)  <X— </)  +  (/X-/)  (/-/)  =  r  0-r')  (A), 


and  we  learn  that  if  x'y'  satisfies  either  (A)  or  (B),  the  tangent  of  S 

will  touch  8'.     Since  (A)  arose  from  the  supposition  &=  0,  that  is, 

from  the  supposition  that  the  radii  of  contact  in  the  two  circles  were 

parallel,   and   lay 

in  the  same  direc- 

tion,   a  moment's 

inspection  of  the 

diagram  will  show 

that,  when  (A)  is 

satisfied,  the  com- 

mon   tangent    is 

direct,     as     MN  ; 

while  as  (B)  arose 

from  fl'  =  TT.  -f  0, 

that  is,  from  sup- 

posing the  radii  of  contact  parallel,  but  lying  in  opposite  directions, 

the  corresponding  common  tangent  must  be  transverse,  as  mn. 

To  find,  then,  the  required  points  on  8,  at  which  if  a  tangent  be 
drawn,  it  will  also  touch  /S",  we  merely  eliminate  between  (8)  and 
either  (A)  or  (B).  Now  the  result  of  this  elimination  will  be  a  quad- 
ratic; hence,  there  are  in  all  four  tangents  common  to  8  and  8': 
two  direct,  and  two  transverse. 


The  Chords  of  Contact.  —  Since  the  points  of  contact, 
Jf  and  M',  as  we  have  just  seen,  both  satisfy  the  condition  (A),  it 
follows  that 

(g/-9}(x-g}  +  (f/-n(y-f}  =  r(r-r>}      (i), 

is  the  equation  to  the  chord  of  contact  for  the  direct  tangents.     Similarly, 
(</-g}(x-g}  +  (f'-f)(y-n----r(r  +  r')         (2), 

is  the  equation  to  mm7,  the  chord  of  contact  for  the  transverse  tan  gents. 
Corollary.  —  If  the  origin  be  transferred  to  the  center  of  8,  g  and/ 
will  vanish,  and  the  chords  of  contact  will  be  represented  by 


If  the  axes  be  now  revolved  until  the  line  of  centers  becomes  the 


CENTERS  OF  SIMILITUDE.  289 


axis  of  x,  f/  will  vanish,  and  the  equation  assume  the  form  x  = 
constant.  Hence,  The  chords  of  contact  corresponding  to  common- 
tangents  of  two  circles  are  perpendicular  to  the  line  of  their  centers. 

341*    Segments    formed    on    the    Line    of   Centers.  —  The 

points  O  and  €/,  in  which  the  two  direct  and  two  transverse 
common  tangents  have  their  respective  intersections,  are  the  poles 
of  (1)  and  (2)  in  Art.  340.  In  these,  if  we  multiply  by  r,  and  then 
divide  by  r  —  ?y  and  r-f?''  respectively,  the  co-efficients  of  x  and  y 
(Art.  305,  Cor.)  will  be  the  co-ordinates  of  O  and  C/,  diminished 
by  the  co-ordinates  of  the  center  of  8.  We  thus  get 


„,     a  =         -  . 


„,     f--          . 

y—j  —  r^r,        y-    r^s 

Now  (Art.  95)  these  values  satisfy  the  equation  to  the  line  of  centers, 
and  show  (Art,  52)  that  x/y/  divides  the  distance  between  gf  and 
ff/f/  in  the  ratio  of  r  and  •/.  Hence,  Common  tangents  of  two  circles 
intersect  on  the  line  of  their  centers,  and  divide  the  distance  between 
those  centers  in  the  ratio  of  the  radii. 

342.  Centers  of  Similitude.  —  If  the  common  tangent  be  made 
the  initial  line,  and  either  O  or  O'  be  taken  for  the  pole,  the  polar 
equation  to  the  circle  iS  (Art.  138)  may  be  written 

reo.(g-a) 


sin  a 


by  merely  substituting  for  d  its  value  r  :  sin  a.    Hence,  for  the  circle  >$', 


_  r  {cos  (0 —  a)  =h  I/cos1'  (ft  —  a)  —  cos2  a} 

sin  « 
Similarly,  for  the  circle  8',  we  get 

r'  { cos  (0  —  a)  ±  l/cosa(0  — a)  —  cos'2 a} 

P2 : • 

sm  a 
Therefore,  pl  :  p2  ::  r  :  r7. 

Now  these  vectors  of  *S'  and  8'  are  the  segments  formed  by  the  two 
circles  on  any  right  line  drawn  through  O  or  €/.  Hence,  All  right 
lines  drawn  through  the  intersection  of  the  common  tangents  of  two  circles 
are  cut  similarly  by  the  circles,  namely,  in  the  ratio  of  the  radii. 

Remark — On  account  of  this  property,  the  points  in  which  the 
common  tangents  intersect  are  called  centers  of  similitude. 


290 


ANALYTIC  GEOMETRY. 


343*    Axis  of  Similitude. — This  name  is  given  to  a  certain 
right  line  whose  relation  to  three  circles  we  will  now  develop. 

Let  gf,  g'f,  g//f//  be  the  centers  of  any  three  circles,  and  r,  r', 
r"  their  radii.    The  co-ordinates  of  the  external  center  of  similitude 
for  the  first  and  second  (Art.  341)  will  then  be 
rg'-Sg 


y  = 


—  r' 


those  of  the  corresponding  center  for  the  second  and  third, 


(2); 


and  those  of  the  corresponding  center  for  the  third  and  first, 
„//_       //  ,,,/y ry// 


y"  = 


(3). 


r"—  r 

Now,  if  we  make  the  necessary  sub- 
stitutions and  reductions,  we  shall  find 
that  (1),  (2),  (3)  satisfy  the  condition 
of  Art.  112.  Hence,  Any  three  homolo- 
gous centers  of  similitude  belonging  to 
three  circles  lie  on  one  right  line,  called 

the  AXIS   OF  SIMILITUDE. 

Corollary. — If  two  circles  touch  each 
other,  one  of  their  centers  of  similitude 
becomes  the  point  of  contact.  Hence, 
If  in  a  group  of  three  circles  the  third 
touches  the  other  two,  the  line  joining 
the  points  of  contact  passes  through  a 
center  of  similitude  of  the  two. 

Remark. — The  homologous  centers  of 
similitude  are  either  all  three  external, 
as  in  the  diagram,  or  else  two  internal 
and  the  third  external.  Corresponding 

to  the  latter  case,  there  will  of  course  be  three  different  axes  of 
similitude;  making  in  all  four  such  axes  for  every  group  of  three 
circles. 

THE  CIRCLE  IN  THE  ABRIDGED  NOTATION. 

344.  We  have  room  for  only  a  few  examples  of  the 
uses  to  which  this  notation  can  be  advantageously  applied 


CIRCLE  IN  ABRIDGED  NOTATION.  291 

in  the  case  of  the  Circle.  The  illustrations  given  will 
afford  the  beginner  some  further  insight  into  the  method, 
and  the  reader  who  desires  fuller  information  must  con- 
sult the  larger  works  to  which  we  have  already  referred 
in  connection  with  this  subject. 

343.  Since  a,  /?,  y  are  the  perpendiculars  dropped  from  any 
point  P  to  the  three  sides  of  a  triangle,  it  is  evident  that  the 
function 

fty  sin  A  -f~  ya  sin  B  -f-  af3  sin  C 

denotes  (Trig.,  874)  the  double  area  of  the  triangle  formed  by 
joining  the  feet  of  those  perpendiculars;  for  the  angle  A,  included 
between  the  sides  ft  and  y,  will  be  either  the  supplement  of  the 
angle  between  the  perpendiculars  ft  and  y,  or  else  equal  to  it:  and 
so,  also,  of  the  angles  B  and  C.  Now  (Art.  236)  if  the  point  P  be 
on  the  circumference  of  the  circumscribed  circle, 
we  have 

ft-y  sin  A  +  ya  sin  B  +  aft  sin  C=  0 ;  /^2l>N 

that  is,  the  triangle  contained  between  the  feet 
of  the  perpendiculars  from  P  vanishes,  and  we 
obtain  the  following  theorem :  The  feet  of  the 
perpendiculars  dropped  from  any  point  in  a  circle 
upon  the  sides  of  an  inscribed  triangle  lie  on  one 
right  line. 

346.  The  equation  to  the  circle  circumscribed  about  a  triangle 
may,  by  factoring,  be  written 

y  (a  sin  B  +  ft  sin  A)  +  aft  sin  C=  0 : 

which  shows  that  the  line  a  sin  B  -f-  ft  sin  A  meets  the  circle  on  the 
line  a,  and  also  on  the  line  ft;  since,  if  a  sin  5+  ft  sin  ^4  =  0  in 
the  above  equation,  we  get  a/3  =  0,  a  condition  satisfied  by  either 
a  =  0  or  ft  =  0.  But  the  only  point  in  which  either  a  or  ft  meets 
the  circle  is  their  intersection:  hence  a  sin  B  +  ft  sin  A  is  the  tangent 
of  the  circle  at  aft.  Now  (Ex.  8,  p.  222)  a  sin  A  +ft  sin  B  is  the  parallel 
to  the  base  of  the  triangle,  passing  through  its  vertex;  and  (Ex.  6, 
p.  222)  this  parallel,  and  therefore  the  base,  has  the  same  inclina- 
tion to  a  or  ft  as  the  tangent  a  sin  B  -f-  ft  sin  A  has  to  ft  or  a.  Hence, 
the  tangent  of  the  circumscribed  circle,  at  the  vertex  of  a  triangle, 


292 


ANALYTIC   GEOMETRY. 


makes  the  same  angle  with  either  side  as  the  base  does  with  the 
other;  or  we  have  the  well-known  theorem:  The  angle  contained  by 
a  tangent  and  chord  of  any  circle  is  equal  to  that  inscribed  under  the 
intercepted  arc. 

347.  The  equation  obtained  in  the  preceding  article  denotes 
the  tangent  at  the  vertex  C  of  a  triangle  inscribed  in  a  circle ;  and 
analogous  equations  may  at  once  be  written  for  the  tangents  at  the 
other  two  vertices  A  and  B.  The  equation  to  the  tangent  at  any 
point  of  a  circle  circumscribed  about  a  given  triangle  (compare 
Arts.  236;  240,  II)  is 


a  sin  A        j3  sin 


y  sin  C 


a'* 


=  0. 


348.  The  equations  to  the  tangents  at 
the  vertices  of  an  inscribed  triangle  (Art. 
346)  may  be  written 


+-B»- 

y 

+  sinC^0 


sin  (7       sin  A 
Now  (Art.  108)  the  line 
ft 


(1), 
(2), 
(3). 


a 
sin.4 


sinB       sinO 


passes  through  the  intersection  of  (1)  with 
7,  of  (2)  with  a,  and  of  (3)  with  /3.  Hence, 
The  tangents  at  the  vertices  of  an  inscribed 
triangle  cut  the  opposite  sides  in  points  which 
lie  on  one  right  line. 

349.   Subtracting  (2)  from  (3)  above,  (3)  from  (1),  and  (1) 
from  (2),  we  get 


_  ___  - 
sin  A       sinj3 


-4- 

s'mB 


-- 

sin  (7 


=0- 
' 


sin  C      s'mA 


which  (Art.  108)  are  the  equations  to  the  three  lines  which  join 
the  intersections  of  the  tangents  at  the  vertices  to  the  intersections 


CIRCLE  IN  ABRIDGED  NOTATION.  293 

of  the  sides.  Hence,  (Art.  114,)  The  lines  which  join  the  vertices 
of  a  triangle  to  those  of  the  triangle  formed  by  drawing  to  its  circum- 
scribed circle  tangents  at  its  vertices,  meet  in  one  point. 

Remark  —  The  theorems  of  the  last  two  articles,  which  are  illus- 
trated in  the  diagram,  are  evidently  a  particular  case  of  homology 
(Art.  327)  due  to  a  pair  of  conjugate  triangles. 

35O.  Radical  Axis  in  Trilinears.  —  The  equations 
to  any  two  circles,  in  the  abridged  notation,  (Arts.  236, 
237)  are 

sin  A       sin  E       sin  C        7ir/,  n 

——  +  -j-  +  —  -  +  M  (la  +  mfi  +  nf)  =  0, 

sin  A      sin  B      sin  C   .    1/f-  /7/  IQ        ,  .       A 

—  —  +  —  g-  -j-  —  —  +  M(l'a  +  m'p  +  n'r)  =  0. 

Hence,  their  radical  axis,  S  —  S',  is  denoted  by 
la  -f  mp  +  nr  =  I'  a  +  m'fi  +  n'r. 

Corollary.  —  The  radical  axis  of  any  circle  and  the 
circle  circumscribed  about  the  triangle  of  reference,  is 
represented  by 

la  -\-  mfi  -{-  nf  =  0. 

EXAMPLES     ON     THE     CIRCLE. 

1.  Find  the  intersections  of  the  line  4.r+3y  =  35c  with  the 
circle 


Also,  the  tangents  from  the  origin  to  the  circle  a;2-f  ?/2—  6s—  2^+8=0. 

I.  Show  that  the  equation  to  any  chord  of  a  circle  may  be 
written 

(X—X')(X-  X")  +  (y  -  y>}  (y  -  y")  =  ^  _|_  yt  _  ^ 


the  origin  being  at  the  center,  and  ary,  x"y"  being  the  extremities 
of  the  chord. 


!    Find  the  polar  of  (4,  5)  with  respect  to  z2+  y2—  3x  —  4y  =  8, 
and  the  pole  of  2x  +  3y  =  6  with  respect  to  (x—  l)2-f  (y—  2)2=  12! 


294  ANALYTIC  GEOMETRY. 

4.  Prove  that  the  condition  upon  which  Ax  -f-  By  +(7=0  will 
touch  the  circle  (x  —  #)2  +  (y  —  /)2  =  r'2  is 

Ag  +  Bf+C  = 
~ 


5.  Find  the  length  of  the  chord  common  to  the  two  circles 

(x  -  «)2  +  (y  ~  P)'2  =  r2,        (x  -  W  +  (y  -  a)2  -  r2. 

Also,  the  equations  to  the  right  lines  which  touch  or  +  y2  =  ?'2  at 
the  two  points  whose  common  abscissa  is  1. 

6.  Find  the  equation  to  the  circle  of  which  y  =  2x-{-3  is  a 
tangent,  the  center  being  taken  for  the  origin. 

1.  Prove  that  the  bisectors  of  all  angles  inscribed  in  the  same 
segment  of  a  circle  pass  through  a  fixed  point  on  the  curve. 

8.  Given  the  hypotenuse  of  a  right  triangle:  the  locus  of  the 
center  of  the  inscribed  circle  is  the  quadrant  of  which  the  given 
hypotenuse  is  the  chord. 

9.  Given  two  sides  and  the  included  angle  of  a  triangle:    to 
find  the  equation  to  the  circumscribed  circle. 

10.  The  locus  of  a  point  from  which  if  lines  be  drawn  to  the 
vertices  of  a  triangle,  their  perpendiculars  through  the  vertices  will 
meet  in  one  point,  is  the  circle  circumscribed  about  the  triangle. 

11.  If  any  chord  be  drawn  through  a  fixed'point  on  the  diameter 
of  a  circle,  and  its  extremities  joined  to  either  end  of  the  diameter, 
the  joining  lines  will  cut  from  the  tangent  at  the  other  end,  portions 
whose  rectangle  is  constant.     [See  Art.  137.] 

12.  The  locus  of  the  intersection  of  tangents  drawn  to  any  circle 
at  the  extremities  of  a  constant  chord  is  a  concentric  circle.     [See 
Art,  307.] 

13.  If  a  chord  of  constant  length  be  inscribed  in  a  given  circle, 
it  will  always  touch  a  concentric  circle. 

14.  If  through  a  fixed  point  O  any  chord  of  a  circle  be  drawn, 
and  OP  be  taken  an  harmonic  mean  between  its  segments  OQ, 
OQf,  the  locus  of  P  will  be  the  polar  of  0. 

15.  If  through  any  point   O  of  a  circle,  any  three  chords  be 
drawn,  and  on  each,  as  a  diameter,  a  circle  be  described,  the  three 
circles  which  thus  meet  in  O  will  meet  in  three  other  points,  lying 
on  one  right  line. 


EXAMPLES  ON  THE  CIRCLE.  295 

16.  If  several  circles  pass  through  two  fixed  points,  their  radical 
axes  with  a  fixed  circle  will  pass  through  a  fixed  point. 

[This  example  may  be  best  solved  by  means  of  the  Abridged  Notation, 
but  can  be  done  very  neatly  without  it.] 

17.  Form  the  equation  to  the  system  of  circles  which  cuts  at 
right  angles  any  system  with  a  common  radical  axis,  and  prove, 
by  means  of  it,  that  every  member  of  the  former  system  passes 
through  the  limiting  points  of  the  latter. 

18.  If  PQ  be  the  diameter  of  a  circle,  the  polar  of  P  with  respect 
to  any  circle  that  cuts  the  first  at  right  angles,  will  pass  through  Q. 

19.  The  square  of  the  tangent  drawn  to  any  circle  from  any  point 
on  another  is  in  a  constant  ratio  to  the  perpendicular  drawn  from 
that  point  to  their  radical  axis. 

20.  If  a  movable  circle  cut  two  fixed  ones  at  constant  angles,  it 
will  cut  at  constant  angles  all  circles  having  the  same  radical  axis 
as  these  two. 

[First  prove  that  the  angle  0  at  which  two  circles  cut  each  other,  is 
determined  by  the  formula 

j)-i  =  jf?2  +  ri  _  ^Rr  cos  0, 

in   which  R,  r  are  the  radii  of  the  circles,  and  D  the  distance  between 
their  centers.] 

21.  Find    the    equations  to   the   common  tangents   of  the   two 
circles 


What  is  the  equation  to  their  radical  axis  ? 

22.  If  a  movable  circle  cut  three  fixed  ones,  the  intersections  of 
the  three  radical  axes  will  move  along  three  fixed  right  lines  which 
meet  in  one  point. 

23.  The  radical  axis  of  any  two  circles  that  do  not  intersect, 
bisects  the  distances  between  the  two  points  of  contact  correspond- 
ing to  each  of  the  four  common  tangents. 

24.  If  through  a  center  of  similitude  belonging  to  any  two  circles, 
we  draw  any  two  right  lines  meeting  the  first  circle  in  the  points 
R  and  Rf  ,  S  and  S/  respectively,  and  the  second  in  r  and  r',  s  and 
s/  :  then  will  the  chords  US  and  rs,  It'S''  and  r's',  be  parallel  ;  while 
US  and  r's',  JR'S'  and  r$,  will  each  intersect  on  the  radical  axis. 

An.  Ge.  28. 


296  ANALYTIC  GEOMETRY. 

25.  Find  the  trilinear  equation  to  the  circle  passing  through  the 
middle  points  of  the  sides  of  any  triangle,  and  prove  that  this  circle 
passes  through  the  feet  of  the  three  perpendiculars  of  the  triangle, 
and  bisects  the  distances  from  the  vertices  to  the  point  in  which 
the  three  perpendiculars  meet.  [This  circle  is  celebrated  in  the 
history  of  geometry,  and,  on  account  of  passing  through  the  points 
just  mentioned,  is  called  the  Nine  Points  Circle.^ 

Find,  also,  the  radical  axis  of  this  and  the  circumscribed  circle. 


CHAPTER   THIRD. 

THE   ELLIPSE, 
i.  THE  CURVE  REFERRED  TO  ITS  AXES. 

We  may  most  conveniently  begin  the  discussion 
of  the  Ellipse  by  means  of  the  equation  which  we  obtained 
in  Art.  147,  namely, 

of    ,    f       i 
a2  +  j* 

At  a  later  point  in  our  investigations,  we  shall  refer  the 
curve  to  lines  which  have  a  relation  to  it  more  generic 
than  that  of  the  two  known  as  the  axes  (Art.  146),  which 
give  rise  to  the  equation  just  written. 

THE   AXES. 

352.  If  in  the  above  equation  we  make  y  =  0,  we 

shall  obtain,  as  the  intercept  of  the  curve  upon  the 
transverse  axis, 

x  =  ±a  (1); 


PROPERTIES  OF  THE  ELLIPSE.  297 

and,  making  x  =  0,  we  get,  for  the  intercept  upon  the 
conjugate  axis, 

y=±b  (2). 

Comparing  (1)  and  (2),  we  see  that  the  curve  cuts  both 
axes  in  two  points,  and  that  in  each  case  these  two  points 
are  equally  distant  from  the  focal  center,  which  (Art.  147) 
was  taken  for  the  origin.  Hence  we  have 

Theorem  I.  —  The  focal  center  of  any  ellipse  bisects  the 
transverse  axis,  and  also  the  conjugate. 

Corollary.  —  We  must  therefore  from  this  time  forward 
interpret  the  constants  a  and  b  in  the  equation 


as  respectively  denoting  half  the  transverse  axis  and 
half  the  conjugate  axis. 

Remark  —  This  theorem  follows,  of  course,  directly  from  that  of 
Art.  149.  We  .have  purposely  developed  it  by  a  separate  analysis, 
however,  in  order  that  the  student  may  see  the  consistency  of  the 
analytic  method. 

353.  If  in  the  equation  of  Art.  147,  which  may  be 
written 

b 


we  suppose  x  >  a  or  <  —  a,  the  corresponding  values 
of  y  are  imaginary;  so  that  no  point  of  the  curve  is 
farther  from  the  origin,  either  to  the  right  or  to  the  left, 
than  the  extremities  of  the  transverse  axis.  Now  (Art. 
147),  for  the  distance  from  the  origin  to  either  focus, 
we  have 


298  ANALYTIC   GEOMETRY. 

Hence,  c  can  not  be  greater  than  a,  though  it  may 
approach  infinitely  near  to  the  value  of  a,  as  b  dimin- 
ishes toward  zero.  Therefore, 

Theorem  II. —  The  foci  of  any  ellipse  fall  within  the 
curve. 

354.  Moreover,  a  —  c  measures  the  distance  of  either 
focus   from  the  adjacent  vertex ;   while  the  distance  of 
either  from   the  remote  vertex  =  a  -f  c.     We   accord- 
ingly get 

Theorem  III. —  The  vertices  of  the   curve   are   equally 
distant  from  the  foci. 

355.  From   Art.  352,  the  length  '  of  the  transverse 
axis  =  2a.     But  (Art.  147)  2a  =  the  constant  sum  of 
the  focal  radii  of  any  point  on  the  curve.     That  is, 

Theorem  IV. — The  sum  of  the  focal  radii  of  any  point 
on  an  ellipse  is  equal  to  the  length  of  its  transverse  axis. 

Corollary. — This  property  gives  rise  to  the  following 
construction  of  the  curve  by  points : 
Divide  the  transverse  axis  at  any 
point  M  between  the  foci  Fr  and  F. 
From  Fr  as  a  center,  with  a  radius 
equal  to  the  segment  MA',  strike  A' 
two  small  arcs,  one  above  the  axis, 
and  the  other  below  it.  Then  from 
F,  with  the  remaining  segment  MA 
as  radius,  strike  two  more  arcs,  intersecting  the  two 
former  in  P  and  P' :  these  points  will  be  upon  the 
required  ellipse  ;  for  F'P  -f  FP=  AA'  =  F'P'  -f  FP'. 
By  using  the  radius  MA1  from  F,  and  MA  from  F',  two 
more  points,  P"  and  P'",  may  be  found ;  so  that  every 
division  of  the  transverse  axis  will  determine  four  points 


CONSTRUCTION  OF  THE  FOCI. 


299 


of  the  curve.  Thus,  the  point  of  division  N  will  give  rise 
to  the  four  points  Q,  Qf,  Q",  Q"f.  When  enough  points 
have  been  found  to  mark  the  outline  of  the  curve  dis- 
tinctly, it  may  be  drawn  through  them  ;  if  necessary, 
with  the  help  of  a  curve-ruler.  It  is  evident  that  this 
construction  implies  that  the  transverse  axis  and  the  foci 
are  given. 

356.  The  abbreviation  b'2~a2  —  c1  adopted  (Art.  147) 
for  the  Ellipse,  gives  us 


b=l(a  +  c)  (a  —  c). 

Hence,  attributing  to  «,  6,  c  the  meanings  now  known  to 
belong  to  them,  we  have 

Theorem  V, — The  conjugate  semi-axis  of  any  ellipse  is 
a  geometric  mean  between  the  segments  formed  upon  the 
transverse  axis  by  either  focus. 

Corollary. — Transposing    in    the 
abbreviation  above,  we  have  b2  +  c2 
=  a2.    But,  from  the  diagram,  b2  -f-  c2 
=  F'B2=FB2.    Therefor<J%g  dis- 
tance from  either  focus  of  an  ellipse 

to  the  vertex  of  the  conjugate  axis  is  equal  to  the  semi- 
transvers&j  We  have,  then,  the  fol- 
lowing construction  for  the  foci, 
when  the  two  axes  are  given:  — 
From  B,  the  vertex  of  the  conju- 
gate axis,  with  a  radius  equal  to 
the  semi-transverse,  describe  an  arc 

cutting  the  transverse  axis  A1  A  in  F1  and  F:  the  two 
points  of  intersection  will  be  the  foci  sought. 

357.  Let  x'y' ,  x"y"  be  any  two  points  of  an  ellipse. 
Then,  from  the  equation  of  Art.  147, 


300 


ANALYTIC   GEOMETRY. 


(i); 


Dividing  the  first  equation  of  (1)  by  the  second,  we  get 
y  »  :  y"*  : :  (a  +  x')  (a  -  x')  :  (a  +  x")  (a  -  x"). 

By  a  like  operation  in  (2),  we  obtain 

x'2 :  x"2  : :  (b  +  y')  (b—y'}  :  (b  +  y")  (b—y"}. 

Now  a  -\-  xf,  a  —  x'  are  evidently  the  segments  formed 
by  y'  upon  the  transverse  axis,  and  a  -\-  x" ,  a  —  x'1  are 
those  formed  by  y".  Similarly,  b  -j-  ?/',  b  —  y'  are  the 
segments  formed  by  x'  upon  the  conjugate  axis,  and 
b  +  y",  b  —  y"  those  formed  by  x".  Hence, 

Theorem  VI. —  The  squares  on  the  ordinates  drawn  to 
either  axis  of  an  ellipse  are  proportional  to  the  rectangles 
under  the  corresponding  segments  of  that  axis. 

Corollary. — If  in  the  first  expression  of  (1)  we  make 
x'  =  ±  6*,  we  get 

b2  .— B 

But  a2—  c2  =  b2.     Hence,  after  re-    A' 
ductions, 

/_  /p/      & 

a 

Now,  either  of  the  double  ordinates  that  pass  through 
the  foci,  PQ  or  P'Q',  is  called  the  lafus  rectum  of  the 
ellipse  to  which  it  belongs.  Hence, 


latus  rectum  =  —  = 
a 


CIRCUMSCRIBED  CIRCLE.  301 

That  is,  The  latus  rectum  of  any  ellipse  is  a  third  propor- 
tional to  the  transverse  axis  and  the  conjugate. 

358.  The  equation  to  the  Ellipse  (Art.  147)  may  be 
thrown  into  either  of  the  forms 

y2  b2  x2  a2 


(a  +  x)(a—x)        a27      (b  +  y)  (b  -  y)  ~  ~  b2 

Hence,  since  a  ratio  is  not  altered  when  both  its  terms 
are  multiplied  by  the  same  number, 

Theorem  VII. — The  squares  on  the  axes  of  any  ellipse 
are  to  each  other  as  the  rectangle  under  any  two  segments 
of  either  is  to  the  square  on  the  ordinate  which  forms  the 
segments. 

Note. — It  may  be  worth  while  to  observe,  in  passing,  that,  in  this 
theorem  and  the  one  of  Art.  357,  the  word  ordinate  has  been  used 
in  aTwider  sense  than  we  originally  assigned  to  it.  We  shall  fre- 
quently have  occasion  to  employ  it  in  this  larger  meaning,  of  a  line 
drawn  to  either  axis  of  co-ordinates  'parallel  to  the  other. 


359.  The  equation  to  the  Ellipse  being  put  into  the 

7,2 0      (~2  T2\  C\\ 

y  -  •  — ,  \a        x )  ^j.j, 


form 


the  equation  to  the  circle  described  on  the  transverse 
axis  as  a  diameter  (Art.  161)  will  be 

f  =  <?-x*  (2). 

Hence,  supposing  the  x  of  (1)  and  (2)  to  become  iden- 
tical, we  get 

ye:  yc::b  :  a. 
That  is  to  say, 

Theorem  VIII. —  The  ordinate  of  any  ellipse  is  to  the 
corresponding  ordinate  of  the  circumscribed  circle,  as  the 
conjugate  semi-axis  is  to  the  semi-transverse. 


302  ANALYTIC  GEOMETRY. 

Corollary  1. — By  similar  reasoning,  we  should  find 
that  the  abscissa  of  an  ellipse  is  to  the  corresponding 
abscissa  of  the  inscribed  circle.,  as  the  transverse  semi-axis 
is  to  the  semi-conjugate.  When  the  axes  are  given,  we 
may  therefore  construct  the  curve  by  either  of  the 
following  methods : 

First:  Describe  circles  upon  the 
given  axes  A! A ,  B'B.  At  any  point 
M  of  the  transverse  axis,  erect  a  per- 
pendicular, and  join  the  point  Q,  in 
which  it  meets  the  outer  circle,  with 
the  common  center  0.  Through  R, 
in  which  QC  cuts  the  inner  circle, 
draw  RP  at  right  angles  to  the  con- 
jugate axis  :  the  point  P,  in  which  RP  cuts  MQ,  will 
be  upon  the  required  ellipse.  For,  by  similar  triangles, 
MP  :  MQ  : :  CR  :  CQ  : :  b  :  a. 

Second  :  Suppose  PS  to  be  a  ruler  whose  length  =  CA. 
From  the  end  P,  lay  off  upon  it  PV '  =  CB.  Set  the  end 
S  against  the  conjugate  axis,  say  at  some  point  below  0, 
and  rest  the  point  V  upon  the  transverse  axis,  say  to  the 
right  of  C.  Move  the  ruler  so  that  $  and  V  may  slide 
along  the  axes  :  the  extremity  P  will  describe  an  ellipse. 
For,  if  MPQ  be  drawn-  through  P  perpendicular  to  CA, 
and  QC  be  joined,  the  latter  will  be  equal  and  thence 
parallel  to  the  ruler  PS,  and  we  shall  have  the  proportion 
MP:  MQ::PV:PS::b:a. 

The  second  of  these  methods  constitutes  the  principle 
of  the  so-called  elliptic  compasses  —  an  instrument  used 
for  describing  ellipses,  and  consisting  of  two  bars,  A' A, 
B'B,  fixed  at  right  angles  to  each  other,  along  which 
a  third,  PS,  slides  freely  upon  two  points,  S  and  V, 
whose  distance  apart  is  constant.  [See  Ex.  6,  p.  167.] 


ECCENTRICITY.     LINEAR  EQUATION. 


303 


Corollary  2. — We   are   now    enabled   to   interpret  the 
abbreviation 

s,1  7,2 


adopted  in  Art.  150.  In  the  first  place,  since  a2  —  b2  =  c2, 
we  learn  that  «_J_s  the  ratio  which  the  distance  from  the 
cgnteZ-  to-  either  focus  of  an  ellipse  bears  to  its  transverse 
semi-axis.  But  (Art.  161)  a  circle  is  an  ellipse  in  which 
£  — 0,  or  in  which,  therefore,  a  =  b.  In  any  circle,  then, 
the  ratio  e  is  equal  to  zero.  Hence,  if  we  compare  ellipses 
having  a  common  transverse  axis  =  2a  with  their  common 
circumscribed  circle,  it  is  evident  (since  e  will  increase  as 
the  conjugate  semi-axis  b  diminishes  from  the  maximum 
value  a  toward  zero)  that  e  may  be  taken  to  measure 
the  deviation  of  any  of  these  ellipses  from  the  circum- 
scribed circle.  For  this  reason,  the  ratio  e  is  called  the 
eccentricity  of  the  ellipse  to  which  it  belongs. 

The  eccentricity  of  any  ellipse  evidently  lies  between 
the  limits  0  and  1.  In  fact  the  name  ellipse  (derived  from 
the  Greek  l/Jsixsev,  to  fall  short)  may  be  taken  as  signi- 
fying, that,  in  this  curve,  the  eccentricity  is  less  than  unity. 

Since  e  increases  as  b  diminishes,  it  is  evident  that  the 
greater  the  eccentricity,  the  flatter  will  be  the  correspond- 
ing ellipse. 

36O.  The  distance  of  any  point  on  an  ellipse  from 
either  focus,  may  be  expressed  in  terms  of  the  abscissa 
of  the  point.  For,  putting  p  to  denote  any  such  focal 
distance,  we  have  (Art.  147) 

p*  =  (c  ±  xf  -f  2/2. 
But,  from  the  equation. to  the  Ellipse, 


»= 


An.  Ge.  29. 


304  ANALYTIC  GEOMETRY. 

Substituting  and  reducing,  and  remembering  (Art.  151) 
that  b2  Jrc2  =  a2,  a2  —  b2  =  a  V,  and  c  =  ae,  we  get 

p  =  a  ±  ex, 

in  which  the  upper  sign  corresponds  to  the  left-hand 
focus,  and  the  lower  sign  to  the  right-hand  one.  Hence, 
since  p  and  x  are  of  the  first  degree, 

Theorem  IX, —  The  focal  radius  of  any  point  on  an 
ellipse  is  a  linear  function  of  the  corresponding  abscissa. 

Remark. — The  expression  just  obtained  is  accordingly 
known  as  the  Linear  Equation  to  the  Ellipse. 

361.  The  form  of  the  Ellipse  is  already  familiar  to 
the  student,  from  the  method  of  generating  it  given  in 
Art.  145.  Its  appearance  shows,  or  at  least  suggests, 
that  it  is  an  oblong,  closed  curve,  continuous  in  extent, 
and  symmetric  to  both  of  its  axes.  But  it  may  be  inter- 
esting, at  this  point,  to  show  how  we  might  have  discovered 
each  of  these  peculiarities  of  form  from  the  equation  itself, 
without  the  aid  of  any  drawing. 

I.  The  curve  is  oblong.    For,  no  matter  where  we  take  it  between 
its  two  limiting  cases,  the  Point  and  the  Circle,  in  its  equation 


b  (since  it  is  equal  to  Va*  —  c2)  must  be  less  than  a;  that  is,  the 
conjugate  axis  must  be  less  than  the  transverse. 

II.  It  is  closed,  i.  e.,  limited  in  the  directions  of  both  axes.  For, 
if  we  suppose  x  >  a  or  <  —  a,  the  corresponding  values  of  y  are 
imaginary;  and,  if  we  suppose  y^>b  or  <  —  b,  the  corresponding 
values  of  x  are  imaginary. 

III.  It  is  continuous  in  extent.     For,  between  the  limits  #  —  — a 
and  x  —  a,  all  the  values  of  y  are  real. 

IV.  It  is  symmetric  to  both  axes.     For,  corresponding   to  every 
value  of  x  between  the  limits  — a  and  a,  the  two  values  of  y  are 
numerically  equal  with  opposite  signs;  and  the  same  is  true  of  the 
values  of  x  corresponding  to  any  value  of  y  between  —  b  and  b. 


BISECTORS  OF  PARALLEL  CHORDS.  305 

DIAMETERS. 

362.  Equation  to  any  Diameter.  —  We    are   re- 

quired to  find  the  equation  to  the  locus  of  the  middle 
points  of  any  system  of  parallel  chords  in  an  ellipse. 
Let  xy  be  the  variable  point  of  this  locus,  6'  the  common 
inclination  of  the  bisected  chords,  and  x'y'  the  point  in 
which  any  chord  of  the  system  cuts  the  curve.  Then 
(Art.  101,  Cor.  3) 

x'  =  x  —  I  cos  6',    y1  =  y  —  I  sin  6'. 
But  x'y'  is  a  point  on  the  curve  ;  hence  (Art.  147) 

(x  —  lcosd1)2    ,    (y  —  Zsin^)2__1 
a*  V 

That  is,  to  determine  the  distance  I  between  xy  and  x'y', 
we  get 

'  +b2xcosd')l 


Now,  xy  being  the  middle  point  of  any  chord,  the  two 
values  of  I  must  be  numerically  equal  with  opposite 
signs.  Therefore  (Alg.,  234,  Prop.  3d)  the  co-efficient 
of  I  vanishes,  and  we  obtain,  as  the  required  equation, 

72 

y  —  --  2  x  cot  0'. 

v  Q> 

Corollary.  —  Let  0  =  the  inclination  of  any  diameter 
to  the  transverse  axis.  The  co-efficient  of  x  in  the 
equation  just  found  is  equal  (Art.  78,  Cor.  1)  to  tan  6. 
Hence,  as  the  condition  connecting  the  inclination  of  any 
diameter  with  that  of  the  chords  which  it  bisects, 

tan  0  tan  0'=  —  -„  . 


306  ANALYTIC  GEOMETRY. 

363.  The  equation  to  any  diameter  conforms  to  the 
type  y  =  mx.     Therefore  (Art.  78,  Cor.  5)  we  have 

Theorem  X.  —  Every  diameter  of  an  ellipse  is  a  right 
line  passing  through  the  center. 

Corollary.  —  Since  6'  in  the  foregoing  condition  is  ar- 
bitrary, 6  is  also  arbitrary.  Hence,  the  converse  of  this 
theorem  is  true  ;  that  is,  Every  right  line  that  passes  through 
the  center  of  an  ellipse  is  a  diameter. 

364.  If  we  eliminate   between   the  equation   to   the 
Ellipse  and  that  of  any  diameter,  the  roots  of  the  result- 
ing equation  will  be 

~ 


which,  it  is  evident,  are  necessarily  real.     Hence, 

Theorem  XI.  —  Every  diameter  of  an  ellipse  cuts  the 
curve  in  two  real  points. 

365.  Length  of  any  Diameter.  —  This  is  of  course 
double  the  radius  vector  given  by  the  central  polar  equa- 
tion (Art.  150),  namely, 


1  —  e2  cos2  6 

Hence,  given  the  inclination  #,  the  length  of  the  corre- 
sponding diameter  can  at  once  be  found. 

366.  From  the  preceding  formula,  it  is  evident  that 
the  diameter  is  longest  when  6  =  0,  and  shortest  when 
6  =  90°.  That  is, 

Theorem  XII.  —  In  every  ellipse,  the  transverse  axis  is  the 
maximum,  and  the  conjugate  axis  the  minimum  diameter. 

Remark.  —  For  this  reason,  the  transverse  axis  is  called 
the  axis  major,  and  the  conjugate,  the  axis  minor. 


CONJ  UGA  TE  DIA  METERS. 


307 


*$G7.  Moreover,  since  6  enters  the  foregoing  formula 
by  the  square  of  its  cosine,  the  value  of  ft  is  the  same  for 
0  and  7i  —  0.  Hence, 

Theorem  XIII.  —  Diameters  which  make  supplemental 
angles  with  the  axis  major  of  an  ellipse  are  equal. 

Corollary.  —  The  converse  of  this  is  also  given  by  the 
formula  ;  so  that,  having  the  curve, 
we  can  always  construct  the  axes  as 
follows:  —  Draw  any  two  pairs  of 
parallel  chords,  and,  by  means  of  A'( 
them,  two  diameters  DQ,  D'C:  their 
intersection  C  (Art.  363)  will  be  the 
center.  From  C  describe  any  circle  cutting  the  curve 
in  four  points  P,  ft  P,  Qf.  The  diameters  PP,  QQ' 
will  then  be  equal,  and,  by  the  converse  of  the  theorem 
above,  the  two  bisectors  of  the  angles  between  them 
will  be  the  axes.  These  bisectors  may  be  drawn  most 
readily  by  forming  the  chords  PQ,  QPf  which  subtend 
the  angles  :  they  will  be  perpendicular  to  each  other 
(Art.  317,  Cor.),  and  their  parallels  through  C  will  be 
the  bisecting  axes  required. 

3GS.  Let  0  and  6'  be  the  inclinations  of  any  two 
diameters  to  the  axis  major.  Then  the  condition  that 
the  first  shall  bisect  chords  parallel  to  the  second  (Art. 

362,  Cor.)  is 

7,2 


But  this  is  also   the  condition  upon  which  the   second 
would  bisect  chords  parallel  to  the  first.     Hence, 

Theorem  XIV.  —  If  one  diameter  of  an  ellipse  bisects 
chords  parallel  to  a  second,  the  second  bisects  chords 
parallel  to  the  first. 


308  ANALYTIC  GEOMETRY. 

369.  Two    diameters   of  an   ellipse   which   are   thus 
related,  are  called   conjugate  diameters,  as  in  the  case 
of  the   Circle.     The   re-appearance  of  this  relation   in 
connection   with   the   Ellipse,   gives   occasion   to   define 
what  is  meant  by  ordinates  to  a  diameter. 

Definition.  —  The  Ordinates  to  a  Diameter  are  the 

right  lines  drawn  from  the  curve  to  such  diameter,  parallel 
to  its  conjugate;  or,  they  are  the  halves  of  the  chords 
which  the  diameter  bisects. 

Corollary,  —  Hence,  To  construct  a  pair  of  conjugate 
diameters,  draw  any  diameter  D'D, 
and  any  two  chords  MN,  PQ  par- 
allel to  it.  Join  the  middle  points 
of  the  latter  by  the  line  S'S,  which 
will  be  the  required  conjugate.  When 
the  center  C  is  not  given,  the  con- 
struction is  effected  by  drawing  S'S  through  the  middle 
of  D'D,  parallel  to  the  two  chords  (double  ordinates  to 
D'D]  by  the  aid  of  which  this  first  diameter  must  in 
such  a  case  be  determined. 

370.  Equation  of  Condition  for  Conjugate  Di- 
ameters. —  This,  as  we  have  already  seen  (Art.  368),  is 

7,2 


371.  This  condition,  since  it  shows  that  the  tangents 
of  inclination  belonging  to  any  two  conjugate  diameters 
have  opposite  signs,  indicates  that  one  of  two  conjugates 
makes  an  acute  angle  with  the  axis  major,  and  the  other 
an  obtuse.  Now  the  axis  minor  makes  a  right  angle  with 
the  axis  major;  hence, 

Theorem  XV.  —  Conjugate  diameters  of  an  ellipse  lie  on 
opposite  sides  of  the  axis  minor. 


EXTREMITIES  OF  CONJUGATES.  309 

372.  Equation  to  a  Diameter  conjugate  to  a 
Fixed  Point. — For  brevity,  we  shall  say  that  a  diam- 
eter is  conjugate  to  a  fixed  point,  when  it  is  conjugate  to 
the  diameter  drawn  through  such  point.  If,  now,  x'y' 
be  any  fixed  point,  the  diameter  drawn  through  it  (Art. 
95,  Cor.  2)  will  be 

y'x-x'y  =  Q  (I). 

The  equation  to  the  conjugate  will  be  of  the  form 

y  =  x  tan  6'  (2), 

in  which  (Art.  370)  tan  6f  is  determined  by  the  condition 

tan  0  tan  0'  =  —  -  . 

a2 

But  (1),  tan#  =y' :  x'.     Hence,  the  equation  sought  is 


Corollary, — The  equation  to  the  diameter  conjugate  to 
that  which  passes  through  the  point  (a,  0)  is  evidently 
#  — 0.  But  the  diameter  through  (#,  0)  is  the  axis  major, 
while  x=Q  denotes  the  axis  minor.  Hence,  The  axes  of 
an  ellipse  constitute  a  case  of  conjugate  diameters. 

It  is  from  this  fact,  that  the  axis  minor  derives  its 
name  of  the  conjugate  axis. 

373*  Problem. — Given  the  co-ordinates  of  the  extremity 
of  a  diameter,  to  find  those  of  the  extremity  of  its  conjugate. 

Let  x'yr  be  the  extremity  of  the  given  diameter.  The 
required  co-ordinates,  found  by  eliminating  between  the 
equation  of  Art.  372  and  that  of  the  Ellipse,  are 

.    ay'  bxf 

*<=±T>    *:T*T' 

374.  The  expressions  just  obtained,  transformed  into  the  pro- 
portions 

xc :  if  : :  a  :  b,     xf :  yc  : :  a :  6, 
give  us 


310  ANALYTIC  GEOMETRY. 

Theorem  XVI  —  The  abscissa  of  the  extremity  of  any  diameter  is  to 
the  ordinate  of  the  extremity  of  its  conjugate,  as  tJie  axis  major  is  to 
the  axis  minor. 

375.  Squaring  the  second  expression  of  Art.  373, 
adding  yf27  and  remembering  that  x'y'  satisfies  the  equa- 
tion to  the  Ellipse,  we  find 


Hence,  in  ordinary  language,  we  have 

Theorem  XVII.  —  The  sum  of  the  squares  on  the  ordinates 
of  the  extremities  of  conjugate  diameters  is  constant,  and 
equal  to  the  square  on  the  semi-axis  minor. 

Remark.  —  The  student  may  prove  the  analogous  prop- 
erty; —  The  sum  of  the  squares  on  the  abscissas  of  the 
extremities  of  conjugate  diameters  is  constant^  and  equal 
to  the  square  on  the  semi-axis  major. 

S76.  Problem.  —  To  find  the  length  of  a  diameter  in 
terms  of  the  abscissa  of  the  extremity  of  its  conjugate. 

Let  a!  be  half  the  length  required.  Then,  x'y9  being  the 
extremity  of  a1  ',  we  have  (Art.  51,  1,  Cor.  2)  a'2  =  x'2  -f  y'2. 
Substituting  for  xr  and  y'  from  Art.  373,  we  get 


But  xc  and  yc  satisfy  the  equation  to  the  Ellipse ;  hence, 
nn  —  (at  _  r  2\   i    &   r  2 ~2       a        P   „  2 

^ )  +  -2  ^c  -  -jr-  *c  • 

Therefore  (Art.  151),  for  determining  the  required  length, 
we  have 

a'2  =  a2  —  e2xc2. 

By  a  precisely  similar  analysis,  we  should  find 
bf2  —a2  —  ezx'2. 


INTERCEPT  ON  FOCAL  RADIUS. 


311 


Comparing  the  expressions  last  found  with  the 
formula  of  Art.  360,  we  obtain 

Theorem  XVIII. —  The  square  on  any  semi-diameter  of 
an  ellipse  is  equal  to  the  rectangle  under  the  focal  radii 
drawn  to  the  extremity  of  its  conjugate. 

378.  The  result  of  Art.  376  leads  to  a  noticeable  property  of 
the  Ellipse,  which  we  may  as  well  develop  in  passing. 

Let  x/y/  be  the  extremity  D  of  any  diameter  DJy '.  The  equation 
to  the  conjugate  diameter  S'S  (Art.  372)  will  be 

The  equation  to  Z>F,  which  joins  D  to  the 
focus,  (Arts.  95,  151)  may  be  written 

Eliminating  y  between  (1)  and  (2),  we  obtain 

(a2/2  +  I V2  —  I*  ae  x')  x  =  a*e  y'2 ; 

or,  since  x/  and  y/  satisfy  the  equation  to  the  Ellipse, 
(a2/-2  -  V  ae  x'}  x  =  V  ae  (a2  —  a/2). 

Hence,  for  the  co-ordinates  of  M,  we  have 


_ 

' 


a-ex*     '          ~        a-ex/ 
The  length  of  DM  (Art.  51,1,  Cor.  1)  will  therefore  be  found  from 


f_~v 

(a~~  ex 


: 


Reducing  the  last  expression,  we  obtain 


,2  _ 


2ae  x 


Now  x^+y/'i=a/2^   (Art.   376)   a2  —  eV  =  (Art.  375,  Rem.) 
a2  —  e~  (a2  —  x''2).     Hence, 

6  =  DM=a: 
a  relation  which  may  be  expressed  by 

Theorem  XIX  —  The  distance  from  the  extremity  of  any  diameter  to 
its  conjugate,  measured  upon  the  corresponding  focal  radius,  is  constant, 
and  equal  to  the  semi-axis  major. 


312  ANALYTIC  GEOMETRY. 

3719.  Let  I)'  denote  the  length  of  the  semi-diameter 
conjugate  to  that  whose  extremity  is  xfyf.  Then  (Art. 
51,  I,  Cor.  2)  we  shall  have 

V  =  x*  +  y.'=  (Art.  147)  jiM-jJ  (a2-*;-). 

Reducing,  and  applying  the  abbreviation  of  Art.  150, 

we  get 

b'2  =  b2+e2x2. 

Now  (Art.  376)  a'2  =  a2  —  e2xc2.     Adding  this  expression 
to  the  preceding,  we  find 

a'2  -j-  b'2  =  a2  +  b\ 

giving  us  the  following  important  property  : 

Theorem  XX.  —  The  sum  of  the  squares  on  any  two  con- 
jugate diameters  of  an  ellipse  is  constant,  and  equal  to  the 
sum  of  the  squares  on  the  axes. 

3SO.  Angle  between  two  Conjugates.  —  Let  <p  de- 

note the  angle  required.    We  shall  then  have  <f>  =  dr  —  0; 
whence  (Trig.,  845,  in) 

sin  <p  =  sin  6'  cos  6  —  cos  6'  sin  6. 

But,  putting  af,  b'  for  the  lengths  of  the  semi-diameters, 
and  x'yf,  xl.yc  for  their  extremities,  we  have 

sin  6'=  yc  :  bf,      cos  0'=  xc  :  b'  ; 
cos  0  =  x'  :  a',      sin  6  =  y'  :  a'. 

TT  xfii,,  —  y'xr 

Hence,  sin  <p  =  -         f  -  ; 

a  o 

or,  substituting  for  xc  and  yc  from  Art.  373,  and  reducing, 


sin  <p  =  - 

ab.a'b' 


PARALLELOGRAM  OF  TWO  CONJUGATES.      313 

Now  x'y'  being  on  the  curve,  (Art.  147)  b2xf2  -f-  a2yf2  =  a2b2. 
Therefore, 

ab 


:-~ Q 


>£J 


381.  The  expression  just  found,  by  a  single  trans- 
formation gives  the  relation 

a'bf  sin  <p  =  ab. 

Now  it  is  evident  from  the 
diagram,  in  which  CD  —  a', 
and  CS=br,  that  the  first  mem- 
ber of  this  equation  denotes  the 
parallelogram  CDRS;  and  the 
second,  the  rectangle  CAQB.  Hence, 

Theorem  XXI.  —  The  parallelogram  under  any  two  con- 
jugate diameters  is  constant,  and  equal  to  the  rectangle 
under  the  axes. 

Remark  —  We  have  drawn  the  parallelogram  and  rectangle  in 
question  as  circumscribed.  Future  investigations  will  justify  the 
figure.  The  property  last  obtained  may  be  otherwise  stated:  The 

triangle  formed  by  joining  the  extremities  of  any  two  conjugate  diameters 
is  constant,  and  equal  to  that  included  behveen  the  semi-axes. 

Corollary  1.  —  If  we  suppose  <p  =  90°,  then  sin  <p  =  1  ; 
and  we  obtain 

a'V  =  ab. 
But  (Art.  379), 


Combining   these    equations,  we    get,    after   the   proper 
reductions, 


That  is 


=  b. 


In  other  words,  In  any  ellipse  there  is  but  one  pair  of  conju- 
gate diameters  at  right  angles  to  each  other,  namely,  the  axes. 


314  ANALYTIC  GEOMETRY. 

Corollary  2. — From  the  formula  (Art.  380),  to  is  obvi- 
ously greatest  *  when  a'b'  is  greatest.  But  since  a'2  -\-  b'1 
is  constant,  the  rectangle  a'b'  has  a  constant  diagonal, 
and  is  therefore  greatest  when  a'  =  b'.  Hence,  The 
inclination  of  two  conjugate  diameters  is  greatest  when 
the  diameters  are  of  equal  lengths. 

382.  The  diameters  corresponding  to  the  condition 
af  =  bf  may  be  appropriately  termed  the  equi-conjug ate 
diameters  of  the  ellipse  to  which  they  belong.  Now 
(Art.  367,  Cor.)  for  the  case  of  equi-conjugates, 

tan  6'  =  —  tan  6 ; 

hence  (Art.  370),  for  the  inclinations  of  the  equi-conju- 
gates to  the  axis  major, 

tan  0  =  ±  -  . 
a 

A' 

If  we   form   the  rectangle    of  the 

axes,  LMNR,  it  is  evident  that  the 

first  of  the  ^ two  values  just  found, 

corresponds  to  the  angle  ACL;  and  the  second,  to  the 

angle  ACM.     Hence, 

Theorem  XXII. —  The  equi-conjugates  of  an  ellipse  are 
the  diagonals  of  the  rectangle  contained  under  its  axes. 

Corollary. — It  follows  directly  from  this,  that  an  ellipse 
can  have  but  one  pair  of  equi-conjugates.  In  this  case, 
(Art.  379,)  2a'2  =  a2+V2;  so  that  (Art.  380) 


sin  <p  — 


.7 
a-  -(-  b2 


383.  We  shall  find  hereafter  that  the  two  lines  just 
brought  to  our  notice  have  a  striking  significance  with 


*  The  angle  <p  is  supposed  to  be  that  angle   between   two   conjugates 
which  is  not  acute. 


TANGENT  OF  THE  ELLIPSE.  315 

respect  to  the  analogy  between  the  Ellipse  and  the 
Hyperbola.  They  in  fact  foreshadow  the  two  remarka- 
ble lines  known  as  the  asymptotes  of  the  latter  curve, 
which,  though  still  the  diagonals  of  the  rectangle  formed 
upon  its  axes,  meet  it  only  at  infinity. 

THE    TANGENT. 

384.  Equation  to  any  Chord. — Let  z'yf,  x"y"  be 
the  extremities  of  any  chord  in  an  ellipse :  then  (Art.  147) 

b*xf2  +  «y2 = i  v2  +  ay /2. 

Hence,  after  transposing  and  factoring, 

1trr 7/  7,2     nr'-L-r" 

y_ y_ _£_    x  \x  /-i  \ 

x"—  x'~          d1' y'+y" 

Now  the  equation  to  the  chord  (Art.  95)  must  be  of  the 
form 

y  —  y'  _  y"—yr 

x  —  x'~~  x"—x'' 

Substituting  from  (1)  for  the  second  member,  the  required 
equation  is 

y-y'  =  __  V m  x'  +  x"  \ 

x  —  x'  dz'yfjry"' 

385.  Equation   to  the   Tangent.  —  Suppose   the 
two  points  in  which  the  chord  cuts  the  curve  to  become 
coincident :   then,  in  the  preceding  expression,  x"  =  xr, 
y" -=  y' \    and  the  required  equation  to  the  tangent,  in 
terms  of  the  point  of  contact  x'y1 ,  is 

y  —  y'  b2    xf . 

x  —  x'  "      ~  tf  'y' ; 

or,  by  clearing  of  fractions  and  remembering  that  x'  and 
y'  satisfy  the  equation  to  the  curve, 

x'x  ,  y'y     -, 
~~   ~    z=1* 


316  ANALYTIC  GEOMETRY. 

386.  Condition  that  a  Right  Line  shall  touch 

an  Ellipse. — Eliminating  y  between  the  line  y=mx-+-n 
and  the  ellipse  ,2 

a2 

we  obtain,  as  the  equation  determining  the  intersections 
of  the  line  and  the  curve,  the  quadratic 

(m2a2  -f  b2)  x2  +  2ma2nx  +  a2  (n2  —  b2)  =  0. 
The  condition  that  this  may  have  equal  roots  is 
(ma2n)2  =  a2  (n2  —  b2)  (m2a2  +'  b2). 

Hence,  after  the  necessary  reductions,  the  required  con- 
dition of  tangency  is 

n  =  i/raV  -j-  b2. 

Corollary. — Every  right  line,  therefore,  whose  equation 
is  of  the  form 

y  =  mx  -f  \/m2a2  -f-  b2 

is  a  tangent  to  the  ellipse  whose  semi-axes  are  a  and  b. 
This  expression,  like  the  corresponding  one  belonging  to 
the  Circle  (Art.  306,  Cor.),  affords  singularly  rapid  solu- 
tions of  problems  which  do  not  involve  the  point  of  contact ; 
and  for  this  reason  is  called  the  Magical  Equation  to  the 
Tangent. 


The  Eccentric  Angle. — If  the  ordinate  of  any  point  P 
on  an  ellipse  be  produced  to  meet  the  cir- 
cumscribed circle  in  Q,  and  Q  be  joined  to 
the  center  C\  the  angle  QCM  is  called  the 
eccentric  angle  of  the  point  P.     We  intro- 
duce it  here,  because  it  serves  the  impor-  A'| 
tant  purpose  of  expressing  the  position  of 
any  point  on  an  ellipse  in  terms  of  a  single 
variable :   a  purpose   sometimes   especially 
useful  in  connection  with  the  equation  to  a  chord  or  a  tangent. 
The  eccentric  angle  is  usually  denoted  by  <j>. 


ECCENTRIC  ANGLE.  317 

It  is  evident  from  the  diagram  that  CM==  CQcosQCM,  and 
MP  =  PR  sin  PRM=  CQ  (MP  :  MQ)  sin  QCJIf  =  (Art.  359) 
CB  sin  QCM.  That  is,  if  x'y'  be  any  point  P  of  an  ellipse, 

x/  =  a  cos  0,      3/x  =  i  sin  0. 

By  means  of  this  relation,  we  can  always  express  a  point  on  an 
ellipse  in  terms  of  the  single  variable  £».  Thus,  the  equation  to  the 
tangent  at  x'y*  becomes 

-  COS0+  V  sin  6  =  I, 
a  b 

388.  Problem.  —  If  a  tangent  to  an  ellipse  passes  through 
a  fixed  pointy  to  find  the  co-ordinates  of  contact. 

Let  x'y'  be  the  required  point  of  contact,  and  x"y"  the 
given  point.  Then,  since  x"y"  must  satisfy  the  equation 
to  the  tangent,  and  x'y'  the  equation  to  the  curve,  we 
shall  have  the  two  conditions 

x'x"        y'y"  _^       x*        y* 
~r        ~  ™      -        ~  ~      '- 


Solving  these  for  x'  and  y',  we  find 


»  \At     V     »*X  .^L_     V*/MfV*V  JVCU  Cb     0 

X  =~  b2xm  4-  aV/T~  "  ' 


Corollary. — From  these  values  it  appears,  that  from 
any  given  point  tivo  tangents  can  be  drawn  to  an  ellipse : 
real  when  b2x"2  +  a?yn<2  >  a2b2,  that  is,  when  the  point  is 
without  the  curve ;  coincident  when  b2xff2  -f-  a2y"2  =  a2b2, 
that  is,  when  the  point  is  on  the  curve ;  imaginary  when 
b2x"2  -f  a?y"2  <  «262,  that  is,  when  the  point  is  within  the 
curve. 


318 


ANALYTIC  GEOMETRY. 


389o  The  equation  to  the  tangent  at  x'yf  (Art.  385)  is 
x'x        y'lf 

—  -f  ir  =  i  C1 ) ; 

a"  or 

and  the  equation  to  the  diameter  conjugate  to  that  whose 
extremity  is  x'y'  (Art.  372)  is 

x  x  i  y  y '    A  /o\ 

~tf      ~¥  ~~  ^  '" 

Now  (Art.  98,  Cor.)  the  lines  (1)  and  (2)  are  parallel. 
That  is, 

Theorem  XXIII. — The  tangent  at  the  extremity  of  any 
diameter  of  an  ellipse  is  parallel  to  the  conjugate  diameter. 

Corollary. — If  we  replace  x'  and  y'  by  — x1  and  — yf9 
equations  (1)  and  (2)  still  satisfy  the  condition  of  paral- 
lelism. Hence,  Tangents  at  the  extremities  of  a  diameter 
are  parallel  to  each  other. 

Remark. — If  the  student  will  form  the  equation  to  the 
parallel  of  (2)  passing  through 
x'y'9  he  will  find  that  it  is  (1). 
In  other  words,  the  converse  of 
our  theorem  is  also  true,  and  we 
can  always  construct  a  tangent 
at  any  point  JP9  by  drawing  the 
diameter  PD  and  its  conjugate, 
and  making  LPM  parallel  to  the 
latter.  In  this  way  we  can  form 
the  circumscribed  parallelogram 
corresponding  to  any  two  diameters  PD,  QD';  and  we 
here  find  the  promised  justification  of  the  statement 
(Art.  381,  Rem.),  that  the  parallelogram  under  two 
conjugates  is  circumscribed,  since  its  sides  must  be 
parallel  to  the  conjugates,  and  therefore  be  tangents 
to  the  curve  at  their  extremities. 


DIRECTION  OF  TANGENT.  319 

39O.  Let  PT  be  a  tangent  to  an  ellipse  at  any  point 
P,  and  let  FP,  F'P  be  the  focal  radii  of  contact.  The 
equations  to  these  lines  may  be 
written  (Arts.  385,  95) 

'^£-*'  <jn  ^-""^- 

yt(XJL-C\ /2/_l_£\  y  0      (F' P\ 

Applying  the  formula  for  the  angle  between  two  lines 
(Art.  96,  Cor.  1),  we  get 


l 


cy'(a2—cx')  cy 


_ 
c)  ~       cy'(d*+cx')  ~        cy'  ' 

Hence,  FPT=ISO°  —  F'PT  =  QPT;  and  we  obtain 

Theorem  XXIV.  —  The  tangent  of  an  ellipse  bisects  the 
external  angle  between  the  focal  radii  drawn  to  the  point 
of  contact. 

Corollary  1.  —  We  therefore  have  the  following  solution 
of  the  problem  :  To  construct  a  tangent  to  an  ellipse  at  a 
given  point.  Through  the  given  point  P,  draw  the  focal 
radii  FP,  F'P,  and  produce  the  latter  until  PQ=FP. 
Join  QF,  and  draw  SPT  perpendicular  to  it  :  SPT  will 
be  the  required  tangent.  For  the  construction  gives  us 
FPT  =  QPT,  according  to  the  theorem  that  the  perpen- 
dicular from,  the  vertex  to  the  base  of  an  isosceles  triangle 
bisects  the  vertical  angle. 

Corollary  2.  —  Since  in  Optics  the  angle  of  reflection 
is  equal  to  the  angle  of  incidence,  while  FPT=  QPT= 
SPF',  all  rays  emanating  from  F  and  striking  the  curve 
will  be  reflected  to  F'\  and  reciprocally.  Hence  it  is, 
that  the  two  points  F,  F1  are  called  the  foci,  or  burning 
points,  of  the  curve. 
An.  Ge.  30. 


320 


ANALYTIC  GEOMETRY. 


391.  If  we  make  y  =  0  in  the  equation  to  the  tangent, 
namely,  in 

b2xfx  -f  a?yfy  =  a?b2, 

we  shall  obtain,  as  the  intercept  of  the  tangent  on  the 
axis  major, 


This  intercept  being  thus  a  third 
proportional  to  the  abscissa  of 
contact  and  the  semi-axis  major, 
we  have  the  following  construc- 
tions : 

I.  To  draw  a  tangent  at  any  point  P  of  the  curve. 
On  the  axis  major,  take  C  T  a  third  proportional  (Art. 
10,  I,  2d)  to  the  abscissa  of  contact  CM  and  the  semi- 
axis  CA.     Join  PT,  which  will  be  the  tangent  required. 

II.  To  draw  a  tangent  from  any  point  T  of  the  axis 
major.     Take  CM  a  third  proportional  to  CT  and  CA, 
and  at  M  draw  the  ordinate  MP\   its  extremity  P  will 
be  the  point  of  contact.     Join  TP,  which  will  be  the 
tangent  sought. 

392.  The  Subtan  gent.—  The  portion  MT  of  the 
axis  major,  included  between  the  foot  of  the  tangent 
and  the  foot  of  the  ordinate  of  contact,  is  called  the 
subtangent  of  the  curve,  to  distinguish  it  from  the  sub- 
tangent  formed  on  any  other  diameter.  For  its  length, 
we  have  MT=CT—CM;  that  is  (Art.  391), 


subtan= 


2—'2 


x' 


Now  a-\-x'=A'M,  and  a — x'  =  MA;  so  that  we  get 


CENTRAL  PERPENDICULAR  ON  TANGENT.     321 


Theorem  XXV. — The  suUangent  of  an  ellipse  is  a  fourth 
proportional  to  the  abscissa  of  contact  and  the  two  segments 
into  which  the  ordinate  of  contact  divides  the  axis  major. 

Corollary. — It  appears  from  the  formula  just  found, 
that  the  subtangent  is  independent   of  b.     Hence,  All 
ellipses  described  upon  a  common  axis  major  will  have  a 
common  subtangent  for  any 
given  abscissa  of  contact.  We 
thus  get  a  construction  of 
the    tangent   by  means   of 
the     circumscribed     circle. 
For,  circumscribe  the  circle 
A  QA ;  and  at  Q,  where  the 

prolonged  ordinate  of  any  point  P  of  the  ellipse  meets 
the  circle,  draw  the  tangent  QT:  then,  by  what  has  just 
been  shown,  T  will  be  the  foot  of  the  tangent  at  P, 
which  may  be  drawn  by  joining  PT. 

If  T,  the  foot  of  the  tangent,  were  given  instead  of  the 
point  of  contact  P,  we  should  draw  TQ  tangent  to  the 
circumscribed  circle,  and,  from  the  point  of  contact  Q, 
let  fall  the  ordinate  QM.  The  point  P  in  which  the 
latter  would  cut  the  ellipse,  would  be  the  required  point 
of  contact ;  and,  on  joining  this  with  the  given  point  T, 
we  should  have  the  tangent  sought. 

393.  Perpendicular  from  the  Center  to  any 
Tangent. — The  length  of  this  is  of  course  the  length 
of  the  perpendicular  from  the  origin  upon  the  line 

b2xfx  -j-  cry'y  =  a2b2. 
Hence,  (Art.  92,  Cor.  2,)  to  determine  it  we  have 

a?b2  ab 

~~l/(a2—e2x'2y 


322  ANALYTIC  GEOMETRY. 

But  (Art.  376)  a2  —  eV  =  bf2.     Hence,  finally, 

ab 
P  =  j,- 

Expressing  this  relation  in  ordinary  language,  and  ob- 
serving the  principle  of  Art.  389,  we  obtain 

Theorem  XXVI.  —  The  central  perpendicular  upon  any 
tangent  of  an  ellipse  is  a  fourth  proportional  to  the  parallel 
semi-diameter  and  the  semi-axes. 

394.  Central  Perpendicular  in  terms  of  its  inclination 
to  the  Axis  Major.  —  For  the  length  of  the  perpendicular  from  the 
origin  upon  the  tangent  whose  equation  (Art.  386,  Cor.)  is 

y  = 
we  have  (Art.  92,  Cor.  2) 


Now   let   6   =  the  inclination  of  the   perpendicular:    then  will 
m  =  —  cot  6,  and  we  get 


p  =  T 

The  following  investigation  will  illustrate  the  usefulness 
of  the  expression  last  obtained,  and  of  the  equation  to  the  tangent 
from  which  it  is  derived. 

Let  it  be  required  to  find  the  locus  of  the  intersection  of  tangents 
to  an  ellipse  which  cut  at  right  angles.  The  inclinations  of  the  two 
tangents  will  be  0  and  90°  +  0;  and  we  shall  have,  for  their  central 
perpendiculars, 


p'2  =  a2sin20  +  62cos20. 


Now,  if  xy  be  the  intersection  of  the  tangents,  the  square  of  its  dis- 
tance from  the  center  will  be  a;2  -f-  y2  =  p2  -\-  p'*.  Hence,  from  what 
has  just  been  proved,  the  co-ordinates  of  intersection  are  connected 
by  the  constant  relation 


FOCAL  PERPENDICULARS  ON  TANGENT.       323 

which  is  the  equation  to  a  circle  concentric  with  the  ellipse,  and 
circumscribed  about  the  rectangle  of  the  axes.*     That  is, 

Theorem  XXVII.  —  The  locus  of  the  intersection  of  tangents  which 
cut  each  other  at  right  angles,  is  the  circle  circumscribed  about  the 
rectangle  formed  on  the  axes. 

396.  Perpendiculars  from  the  Foci  to  any 
Tangent.  —  The  co-ordinates  of  the  right-hand  focus 
are  x  =  ae,  y  =  0  :  hence,  for  the  length  of  the  perpen- 
dicular from  F  upon  the  tangent  b2xfx  -j-  azyty  =  a2b2,  we 
have  (Art.  105,  Cor.  2) 

b2x'ae  —  a2b2  ab2  (exf  —  a)  b  (ex'  —  a) 

~         4f2--2  ~ 


Now  (Art.  360)  a  —  ex'=p,  the  right-hand  focal  radius 
of  contact;  and  (Art.  376)  a2  —  e2x'2  =  b'2,  the  square  of 
the  semi-diameter  conjugate  to  the  point  of  contact. 
Hence,  for  the  right-hand  focus, 


Similarly,  for  the  left-hand  focus,  we  should  find 

,       ¥ 

p  —  •      • 

Corollary.—  By  Art.  377,  b'2  —  pp':  hence,  after  squar- 
ing the  expressions  just  obtained, 

9       b2p          .„      b2.of 

r=y>  P=ir: 

formulse  which,  in  certain   cases,  are  more  useful  than 
the  preceding. 


*  This  locus  may  be  obtained  even  more  readily,  as  follows :  —  The 
equations  to  the  two  tangents  (Arts.  386,  Cor.  ;  99,  Cor.)  may  be  written 

y  —  mx  =  r  m2a2  +  62,       my  +  x  —  VHaMP  +  a'2. 

Squaring  and  adding  these  expressions,  we  eliminate  m,  and  get 

a2  +  52. 


324  ANALYTIC  GEOMETRY. 

SOT.   Comparing  the  two  results  of  Art.  396,  we  get 

p  :  p'  : :  p  :  p', 
a  relation  expressed  in  ordinary  terms  by 

Theorem  XXVIII, — The  focal  perpendiculars  upon  any 
tangent  of  an  ellipse  are  proportional  to  the  adjacent  focal 
radii  of  contact. 

SOS.  Multiplying  together  the  values  of  p  and  p', 
and  observing  Art.  377,  we  obtain 

Pp'=b\ 

which  is  the  algebraic  expression  of 

Theorem  XXIX. —  The  rectangle  under  the  focal  perpen- 
diculars upon  any  tangent  is  constant,  and  equal  to  the 
square  on  the  semi- axis  minor. 

SOO.  The  equation  to  any  tangent  of  an  ellipse 
(Art.  386,  Cor.)  being 


y  —  mx  =  v  ma?  -j-  62, 

that  of  the   focal  perpendicular,  which  passes   through 
(1/V— -  b'\  0),  may  be  written  (Art.  103,  Cor.  2) 


my  -f-  x  —  V  a"  —  b2. 

Squaring   these   equations,   and   adding   them   together, 
we  obtain 


as  the  equation  to  the  locus  of  the  point  in  which  the  focal 
perpendicular  meets  the  tangent.  Hence,  (Art.  136,) 

Theorem  XXX.—  The  locus  of  the  foot  of  the  focal  per- 
pendicular upon  any  tangent  of  an  ellipse,  is  the  circle 
circumscribed  about  the  curve. 

Corollary.  —  From  this  property,  we  obtain  the  follow- 
ing method  of  constructing  a  tangent  to  any  ellipse,  — 


GENERIC    CONSTR  UCTION  FOR  TANGENT.      325 


a  method  which  deserves  special  attention,  because  it  is 
applicable  alike  to  all  the  Conies,  and  holds  good  whether 
the  point  through  which  the  tangent  is  drawn  be  without 
the  curve  or  upon  it. 

To  dratv  a  tangent  to  an  ellipse  through  any  given 
point. — Join  the  given  point  P  with  either  focus  F,  and 
upon  PF  as  a  diameter  describe 
a  semicircle.  Then  through  §, 
where  this  semicircle  cuts  the 
circumscribed  circle,  draw  PQ: 
it  will  touch  the  ellipse  at  some 
point  T.  For  the  angle  FQP  is 
inscribed  in  a  semicircle,  and  Q  is  therefore  the  foot  of 
the  focal  perpendicular  upon  PQ. 

In  case,  as  in  the  second  dia- 
gram annexed,  the  point  P  is  on 
the  ellipse,  the  circle  described 
on  PF  will  be  found  to  touch 
the  circumscribed  circle  at  Q 
(see  Ex.  8,  p.  359).  The  con- 
struction still  holds,  however ;  for  the  point  of  contact  Q 
must  lie  on  the  line  joining  the  centers  of  the  auxiliary 
and  circumscribed  circles,  and  may  therefore  be  found 
at  once  by  joining  C  with  the  middle  point  of  PF,  and 
producing  the  line  thus  formed  until  it  meets  the  cir- 
cumscribed circle. 

Remark.— It  is  obvious  that  the  ordinary  method  of  drawing  a 
tangent  to  a  circle  through  a  given  point  (Geom.,  230),  is  only  a 
particular  case  of  the  method  here  described:  the  case,  namely, 
where  the  two  foci  of  the  ellipse  become  coincident  at  (7,  when  of 
course  the  ellipse  becomes  identical  with  the  circumscribed  circle. 
We  have  seen  (Art.  388,  Cor.)  that  two  tangents  can,  in  general, 
be  drawn  to  an  ellipse  from  a  given  point  P;  and  the  construction 
evidently  corroborates  this,  since  the  auxiliary  circle  PQF  must  in 
general  cut  the  circumscribed  circle  in  two  points. 


326 


ANALYTIC  GEOMETRY. 


4OO.  From  what  has  been  shown  in  the  preceding 
article,  it  follows  that  every  chord  drawn  through  the 
focus  of  an  ellipse  to  meet  the  circumscribed  circle  is  a 
focal  perpendicular  to  some  tangent  of  the  ellipse.  Now 
it  is  evident,  that,  a  circle  being  given,  any  point  within  it 
may  be  considered  as  the  focus  of  an  inscribed  ellipse. 
We  have,  then, 

Theorem  XXXI.  —  If  from  any  point  within  a  circle  a 
chord  be  drawn,  and  a  perpendicular  to  it  at  its  extremity, 
the  perpendicular  will  be  tangent  to  the  inscribed  ellipse  of 
which  the  point  is  .a  focus. 

Corollary  —  This  is  of  course  equivalent  to  saying  that  the  inscribed 
ellipse  is  the  envelope  of  the  perpendicular. 
Advantage  may  be  taken  of  this  principle, 
to  construct  an  ellipse  approximately  by 
means  of  right  lines;  for  it  is  evident  that 
by  taking  the  perpendiculars  sufficiently 
near  together,  we  can  approach  the  line 
of  the  curve  as  closely  as  we  please.  The 
diagram  presents  an  example  of  this  method. 

4Olo  If  the  student  will  now  form,  by  the  method  of 
Art.  108,  Cor.  1,  the  equation 
to  the  diameter  CQ,  that  is, 
the  equation  to  the  line  join- 
ing the  origin  to  the  inter- 
section of  the  tangent 


with  its  focal  perpendicular  (Art.  103,  Cor.  2) 

a2yfx  —  b2x'y  =  a2cy', 
he  will  find  that  it  may  easily  be  reduced  to  the  form 

y'x-(x'  +  c)y  =  0  (CQ). 


POINT  OF  CONTACT.    NORMAL.  327 

Now  the  equation  to  the  focal  radius  of  contact  F'T, 
which  passes  through  the  focus  ( — c,  0)  and  the  point 
of  contact  x'y1 ',  (Art.  95)  may  be  written 

y'x  —  (x>  +  c)  y  +  cy'  =  0  (F* T). 

But  (Art.  98,  Cor.)  these  equations  show  that  CQ  and 
F'T  are  parallel;  and,  by  like  reasoning,  the  same  may 
be  proved  of  CQ'  and  FT.  Hence, 

Theorem  XXXII. — The  diameters  which  pass  through 
the  feet  of  the  focal  perpendiculars  upon  any  tangent  of 
an  ellipse,  are  parallel  to  the  alternate  focal  radii  of 
contact. 

Corollary, — The  equations  to  CQ  and  F'T  evidently 
involve  the  converse  theorem,  Diameters  parallel  to  the 
focal  radii  of  contact  meet  the  tangent  at  the  feet  of  its 
focal  perpendiculars.  Hence,  if  in  drawing  a  tangent 
through  a  given  point  P  it  becomes  desirable,  after 
obtaining  (Art.  399,  Cor.)  the  foot  Q  of  the  focal  per- 
pendicular, to  find  the  point  of  contact,  we  can  do  so 
by  merely  drawing  F'T  parallel  to  CQ. 

By  combining  this  property  with  Arts.  389,  399,  we 
learn  that  the  distance  between  the  foot  of  the  perpen- 
dicular drawn  from  either  focus  to  a  tangent,  and  the  foot 
of  the  perpendicular  drawn  from  the  remaining  focus  to 
the  parallel  tangent,  is  constant,  and  equal  to  the  length 
of  the  axis  major. 

THE    NORMAL. 

4O2.  Equation  to  the  Normal. — The  expression 
for  the  perpendicular  drawn  through  the  point  of  contact 
to  the  tangent 


An.  Ge.  31. 


328  ANALYTIC   GEOMETRY. 

according  to  Art.  103,  Cor.  2,  is 


Clearing  of  fractions,  dividing  through  by  x'y',  and 
putting  (Art.  151)  c2  for  a2  —  52,  we  may  write  the 
equation  sought 


4O3.  If  we  seek  the  angle  tp  made  by  the  normal 
with  the  left-hand  focal  radius  of  contact  F'T,  whose 
equation  is  y'x  —  (x1  -f-  c)  y  -f-  cyf  —  0,  we  get  (Art.  96, 
Cor.  1) 


(xf  +  g) 


Similarly,  for  the  angle  <pf  made  with  the  normal  by  the 
right-hand  focal  radius  of  contact  FT,  wo  get 


,       - 

a:'  _  «/' 

,  y(7^Q=    --F-- 
*f  y1 

Hence  <pr  =  (p  ;  or,  the  normal  makes  equal  angles  with 
the  two  focal  radii  drawn  to  the  point  of  contact,  and 
ve  have 

Theorem  XXXIII.  —  The  normal  of  an  ellipse  bisects  the 
internal  angle  between  the  focal  radii  of  contact. 

Corollary  1.  —  This  property  enables  us  to  construct  a 
normal  at  a  given  point  on  the  curve.     For  let  P  be  the 


CONSTRUCTION  OF  NORMAL. 


329 


given  point,  and  draw  the  corresponding  focal  radii  FP, 
F'P.     On  F'P  lay  off  PQ  equal  to  FP,  join  QF,  and 
draw    PN    perpendicular    to    the 
latter:  PN  (Geom.,  271)  will  bisect 
the  angle  F'PF,  and  will  therefore 
be  the  normal  required. 

Corollary  2. — We  can  also  draw 
a  normal  through  any  point  on  the 

axis  minor.  Let  R  be  such  a  point.  Pass  a  circle 
through  the  given  point  and  the  foci :  it  will  cut  the 
ellipse  in  P  and  P1 '.  Join  R  with  either  of  these  points, 
as  P:  then  will  RP  be  a  normal.  For  the  arc  FfRF 
will  be  bisected  in  R,  and  the  inscribed  angles  F'PR, 
FPR  will  therefore  be  equal ;  that  is,  RP  will  bisect  the 
angle  F'PF. 

4O4.  Intercept  of  the  Normal. — If  in  the  equation 
to  the  normal  (Art.  402)  we  make  y  =  0,  we  find,  as  the 
length  of  the  intercept  on  the  axis  major, 


Corollary.  —  By  means  of 
this  value,  we  can  construct 
a  normal  either  through  a 

given  point  on  the  axis  major  or  at  a  given  point  on  the 
curve.  For,  in  the  former  case,  we  have  CN  given,  to 
find  x'  =  CM\  and,  in  the  latter,  CM  is  given,  to  find 


4O5.  By  Art.  151,  we  have  F'C  =  ae  =  OF.     From 
the  preceding  article,  therefore, 


F'N  =  e  (a  +  ex'\     FN=  e  (a  —  ex1). 


330  ANALYTIC  GEOMETRY. 

Hence  (Art.  360),  F'N  :  FN=F'P  :  FP;  that  is, 

Theorem  XXXIV.  —  The  normal  of  an  ellipse  cuts  the 
distance  between  the  foci  in  segments  proportional  to  the 
adjacent  focal  radii  of  contact. 

406.  length   of  the    Subnormal.  —  The    portion 
NM  of  the  axis  major,  in- 

cluded between  the  foot  of 

the  normal  and  that  of  the 

ordinate  of  contact,  is  called 

the  subnormal  of  the  curve, 

to   distinguish  it  from  that 

formed  on  any  other  diameter.     For  its  length,  we  have 

NM=CM—CN=x'—e2xf=  (l—e2)x'.    That  is  (Art.  151), 

12 

subnor  =  —  xf. 
a" 

407.  Comparing  the  results  of  Arts.  404  and  406, 

ON"  :  NM=  c2  :  b2.    Hence,  as  c2  =  a2  —  b\  we  get 

Theorem  XXXV,  —  The  normal  of  an  ellipse  cuts  the 
abscissa  of  contact  in  the  constant  ratio  (a2  —  b2)  :  b2. 

408.  Length  of  the  Normal.  —  By  this   is   meant 
the  portion  of  the  normal  intercepted  between  the  point 
of  contact  and  either  axis.     We  have,  then, 

PN2  =  PM2  -f  NM2=y'2  -f  b^x'2=  ^  (a2  —  e2x'2). 
Hence,  since  (Art.  376)  a2  —  eW=b'\ 


By  similar  reasoning,  the  details  of  which  are  left  for 
the  student  to  supply, 


SEGMENTS  OF  NORMAL. 


331 


409.  From  the  foregoing,  it  follows  immediately  that 
PN.PR  =  b'2.     In  other  words,  we  have  obtained 

Theorem  XXXVI.  —  The  rectangle  under  the  segments 
formed  by  the  two  axes  upon  the  normal  is  equal  to  the 
square  on  the  semi-diameter  conjugate  to  the  point  of 
contact. 

Corollary.—  We  have  proved  (Art.  377)  that  b'2=pp'. 
Hence,  PN.PH=ppf;  and  we  get  the  additional  property: 
The  rectangle  under  the  segments  of  the  normal  is  equal  to 
the  rectangle  under  the  focal  radii  of  contact. 

410.  It  has  been  shown  (Art.  393)  that,  for  the  length 
of  the  central  perpendicular  upon  any  tangent,  we  have 


. 

Therefore,  CQ.PR  =  a2,  and  CQ.PN=b2.     That  is, 

Theorem  XXXVII.  —  The  rectangle  under  the  normal 
and  the  central  perpendicular  upon  the  corresponding 
tangent  is  constant,  and  equal  to  the  square  on  the 
semi-axis  other  than  the  one  to  which  the  normal  is 

measured. 


SUPPLEMENTAL    AND    FOCAL    CHORDS. 

411.  Definition. — By  Supplemental  Chords  of  an 

ellipse,  are  meant  two  chords  passing 
through  the  opposite  extremities  of 
any  diameter,  and  intersecting  on 
the  curve. 

Thus,  DP,  D'P  are  supplemental 
with  respect  to  the  diameter  D'D\ 
and  A.Q,  A'Q,  with  respect  to  the  axis  major. 


332  ANALYTIC   GEOMETRY. 

412.  Condition  that  Chords  of  an  Ellipse  be 
Supplemental.  —  Let  <p  be  the  inclination  of  any  chord 
DP,  and  <pr  that  of  the  supplemental  chord  D'P.  Then, 
since  (Art.  149)  every  diameter  is  bisected  by  the  center  (7, 
if  the  co-ordinates  of  D  be  a/,  yr,  those  of  D'  will  be  —  x1  ', 
-/;  and  the  equations  to  DP,  D'P  (Art.  101,  Cor.  1) 
may  be  written 

y  —  yf  =  (x  —  x')  tan  y>,      y  -f-  y'  =  (x  -+-  #')  tan  ^'. 

Hence,  at  the  intersection  P,  we  shall  have  the  condition 

y2  —  ?//2  =  (a;2  —  xr2)  tan  ^  tan  <p'9 

in  which  zy,  a/?/',  being  both  upon  the  curve,  are  so  con- 
nected (Art.  147)  that 


The   supplemental  chords  are  therefore  subject  to  the 
constant  condition 

b2 

tan  (p  tan  <p  =  --  -  . 
a2 

413.  If  6  and  6'  are  the  respective  inclinations  of  two 
diameters  drawn  parallel  to  a  pair  of  supplemental  chords, 
then  0=<p  and  6'=<pf  ;  and,  from  the  preceding  condition, 
we  have 

tan  0  tan  6'=  --  . 

a2 

But  this  (Art.  370)  is  the  condition  that  the  diameters 
corresponding  to  6  and  6'  shall  be  conjugate.     Hence, 

Theorem  XXXVIII.  —  Diameters  of  an  ellipse  which  ajre 
parallel  to  supplemental  chords  are  conjugate. 


SUPPLEMENTAL  CHORDS.  333 

Remark. — This  theorem  may  be  otherwise  stated: 
If  a  diameter  be  parallel  to  one  of  two  supplemental 
chords,  its  conjugate  will  be  parallel  to  the  other.  It 
therefore  gives  rise  to  the  following  constructions. 

Corollary  1, — To  construct  a  pair  of  conjugate  diameters 
at  a  given  inclination.  On  any  diameter,  describe  (Geom., 
231)  an  arc  containing  the  given  angle,  and  join  either 
of  the  remaining  points  in  which  the  circle  cuts  the  ellipse 
with  the  extremities  of  the  diameter  :  the  diameters  drawn 
parallel  to  the  supplemental  chords  thus  formed  will  be 
the  conjugates  required. 

Caution. — It  should  be  borne  in  mind,  in  connection  with  this 
problem,  that  the  inclination  of  two  conjugates  in  an  ellipse  is 
subject  to  a  restriction,  and  is  not  any  angle  we  please,  but  only 
(Arts.  381,  Cor.  1 ;  382,  Cor.  2)  any  angle  between  the  limits  90°  and 
sin"1  2ab  :  (a2  +  b2). 

Corollary  2. —  To  construct  a  tangent  parallel  to  a  given 
right  line.  Let  LM  represent  the  given  line.  The  point 
of  contact  of  the  required  tan- 
gent (Art.  389)  is  the  extrem- 
ity of  the  diameter  conjugate 
to  that  drawn  parallel  to  LM. 
Draw,  then,  the  chord  A  Q  par- 
allel to  LM  from  the  extremity 
of  the  axis  major,  and  the  di- 
ameter DP  parallel  to  the  supplemental  chord  QA' ': 
the  line  PT  drawn  through  the  extremity  P  of  this 
diameter,  parallel  to  the  given  line  LM,  will  be  the 
tangent  sought. 

Corollary  3. —  To  construct  the  axes  in  the  empty  curve. 
Draw  any  two  parallel  chords,  bisect  them,  and  form  the 
corresponding  diameter,  say  DP.  On  the  latter,  describe 
a  semicircle  cutting  the  ellipse  in  R.  Join  DJt9  RP: 


334  ANALYTIC   GEOMETRY. 

they  will  be  supplemental  chords  of  the  circle,  and  there- 
fore (Art.  317,  Cor.)  at  right  angles.  They  will  also  be 
supplemental  chords  of  the  ellipse  :  hence  A'A,  J3fJ3, 
drawn  parallel  to  them  through  (7,  will  be  the  rectangular 
conjugates  of  the  curve  ;  that  is,  its  axes. 

414.  Definition.  —  A  Focal  Chord  of  an  ellipse,  and  in 
fact  of  any  conic,  is  simply  a  chord  drawn  through  a  focus. 

The  focal  chords  possess  some  special  properties, 
several  of  which,  in  the  form  corresponding  to  the 
Ellipse,  will  be  given  in  the  examples  at  the  close  of 
this  Chapter.  One  of  them,  however,  has  an  important 
bearing  upon  the  properties  of  a  certain  element  of  the 
curve,  and  we  shall  therefore  develop  it  here. 

415.  The   equation  to   any  focal  chord,  having  the 
inclination   6   to  the  axis  major,  may  be  written  (Art. 
101,  Cor.  3) 

y     =  x  —  ae  __  l 
sin#  ~      costf 

I  being  the  distance  from  the  focus  to  any  point  on  the 
chord.  At  the  intersections  of  the  chord  with  the  curve, 
we  shall  therefore  have  (Art.  147) 

(a2  sin2  6  +  &2  cos2  0)  I2  +  2b2ae  cos  d.l  =  b*. 
The  roots  of  this  quadratic  are  readily  found  to  be 

—  ecosfl) 


~  a  (1  —  e2cos2#)  '  a  (1  —  e2cos-0)  * 

But  these  roots  are  the  values  of  the  two  opposite  seg- 
ments into  which  the  focus  divides  the  chord.  Neglecting, 
then,  the  sign  of  I",  we  have,  for  the  length  of  the  whole 
chord, 

,          2  b2 

cho  =  -  . 


a    1  —  e2  cos2  6 


ELLIPSE  REFERRED  TO  CONJUGATES. 


335 


Now  (Art.  365)  the  second  factor  in  this  expression  is 
the  square  on  the  semi-diameter  whose  inclination  is  0. 
Hence,  if  we  put  a!  =  the  semi-diameter  parallel  to  the 

chord,  we  get 

9/»'2          /9V\2 

i  tM  (—''    I 

cho  =  --  -  = —  : 

a  2a 

a  property  which  we  may  express  by 

Theorem  XXXIX, — Any  focal  chord  of  an  ellipse  is  a 
third  proportional  to  the  axis  major  and  the  diameter 
parallel  to  the  chord. 

Remark, — This  result  is  exemplified  in  the  value  found 
(Art.  357,  Cor.)  for  the  latus  rectum,  which  is  the  focal 
chord  parallel  to  the  axis  minor. 


n.  THE  CURVE  REFERRED  TO  ANY  TWO  CONJUGATES. 

DIAMETRAL  PROPERTIES. 

416.  We  are  now  ready  to  consider  the  Ellipse  from 
a  point  of  view  somewhat  higher  than  the  one  we  have 
hitherto  occupied,  and  shall  presently  discover  that  many 
of  the  properties  we  have  developed  are  only  particular 
cases  of  theorems  more  generic.     Heretofore,  we   have 
referred  the  curve  to  its  axes :  let  us  now  refer  it  to  any 
two  conjugate  diameters. 

417.  Equation  to  the  Ellipse,  referred  to  any 
two  Conjugate  Diameters. — To  obtain  this,  we  must 
transform  the  equation  of  Art.  147, 


336  ANALYTIC  GEOMETRY. 

from  rectangular  axes  to  oblique  ones  having  the  respective 
inclinations  0  and  0'  to  the  axis  major.  Replacing  (Art. 
56,  Cor.  1)  x  and  y  by  x  cos  Q-\-y  cos  0'  and  x  sin  0  -J-  ?/  sin  0', 
we  obtain,  after  obvious  reductions, 

(a2  sin2  0  -f-  62  cos2  0)  x2  -f  (a2  sin2  0'-f  &2  cos2  0')  y* 

-f  2  (a2  sin  0  sin  0'  -f  62  cos  0  cos  0')  #?/  =  a2b2. 

But,  as  the  new  reference-axes  are  conjugate  diameters, 
we  have,  by  a  single  transformation  of  the  condition  in 
Art.  370, 

a2  sin  0  sin  6'  -f  b2  cos  0  cos  0'  =  0. 

The  transformed  equation  is  therefore  in  fact 
(a2  sin2  0  4-  52  cos2  0)  x2  +  (a2  sin2  0'  -f  52  cos2  0')  ?/2  =  a252. 

In  this,  the  co-efficients  are  still  functions  of  the  semi- 
axes  ;  but  if  we  seek  the  values  of  the  semi-conjugates 
a'  and  b1  by  finding  (Art.  73)  the  intercepts  of  the  curve 
upon  the  new  axes  of  reference,  we  readily  obtain 

a2sin20  +  52cos20  =  ^  ,      a2sm26f  -f  62cos20'  =  ^ . 
a'2  bf2 

Hence,  the  required  equation,  in  its  final  form,  is 

a75  +  b"==l' 

4I8»  Comparing  this  equation. with  that  of  Art.  147, 
and  remembering  (Art.  372,  Cor.)  that  the  axes  are  a 
case  of  conjugates,  it  becomes  evident  that  the  equation 
hitherto  used,  namely, 

a2  +  b2  =  1? 


DIAMETRAL  PROPERTIES.  337 

is  only  the  particular  form  assumed  by  the  general  one 
we  have  now  obtained,  when  the  reference-conjugates 
have  the  specific  lengths  2a,  26,  and  are  at  right  angles 
to  each  other.  Moreover,  from  the  identity  of  form  in 
the  two  equations,  we  see  at  once  that  the  transforma- 
tions applied  to 

T2  1J2 

-4-^—1 
a2  +  b2  ~ 

are  equally  applicable  to 


and  that  the  theorems  derived  from  the  former  may 
therefore  be  immediately  extended  to  any  conjugate 
diameters,  provided  they  do  not  involve  the  inclination 
of  the  reference-axes.  Thus,  we  learn  that  the  theorems 
of  Arts.  357,  358  are  particular  cases  of  the  following: 

Theorem  XL.  —  The  squares  on  the  ordinates  to  any 
diameter  of  an  ellipse  are  proportional  to  the  rectangles 
under  the  corresponding  segments  of  the  diameter. 

Theorem  XLI.  —  The  square  on  any  diameter  of  an 
ellipse  is  to  the  square  on  its  conjugate,  as  the  rectangle 
under  any  tivo  segments  of  the  diameter  is  to  the  square 
on  the  corresponding  ordinate. 

410.  The   equation  of  Art.   417  may  of  course   be 

written 


The   equation  to   a  circle  described  upon  the  diameter 
2a'  is 


338  ANALYTIC  GEOMETRY. 

Supposing,  then,  that  we  consider  those  ordinates  of  the 
two  curves  that  correspond  to  a  common  abscissa,  we 
get 

ye:yc  =  V:  a', 

and  therefore  have  the  following  extension  of  the  rela- 
tion (Art.  359)  between  the  Ellipse  and  the  circumscribed 
circle : 

Theorem  XLII. — The  or  dinette  to  any  diameter  of  an 
ellipse  is  to  the  corresponding  ordinate  of  the  circle  de- 
scribed on  that  diameter,  as  the  conjugate  of  the  diameter 
is  to  the  diameter  itself. 

Corollary  1, — Hence,  given  two 
conjugate  diameters  in  position  and 
magnitude,  we  may  construct  the 
curve  by  points,  as  follows:  —  On 
each  of  the  given  diameters  DL, 
D'L',  describe  a  circle.  At  M, 
any  point  on  the  diameter  selected 

for  the  axis  of  x9  set  up  a  rectangular  ordinate  of  the 
corresponding  circle,  meeting  the  curve  in  Q,  and  draw 
MP  parallel  to  the  conjugate  diameter  D'L' .  Join  Q, 
the  extremity  of  the  circular  ordinate,  to  the  center  (7, 
and  through  0,  where  QC  cuts  the  inner  circle,  draw  OR 
parallel  to  DL :  then  will  RM  measure  the  distance  of  0 
from  DL.  From  M  as  a  center,  with  the  radius  MR, 
describe  an  arc  cutting  MP  in  P:  then  will  P  be  a 
point  of  the  required  ellipse.  For  MP  =  MR  -  -  the 
perpendicular  from  0  upon  CD.  Hence,  MP  :  MQ  — 
CO:  CQ  =  bf  :  a'. 

Corollary  2. — The  use  of  the  equation  x2  -f-  y2-  =  a12  in 
the  preceding  investigation,  to  denote  a  circle  described 
upon  a  diameter  of  an  ellipse,  suggests  a  point  of  con- 
siderable importance.  The  equation  denotes  such  a 


ELLIPSE  REFERRED  TO  EQUI-CONJUGATES.    339 

circle,  only  on  the  supposition  that  the  axes  of  refer- 
ence are  rectangular;  and,  in  the  construction  just 
explained,  the  ordinate  MQ  was  drawn  in  accordance 
with  this  principle. 

The  equation,  however,  is  susceptible  of  a  more  gen- 
eral interpretation.  It  evidently  arises  from  the  general 
equation 

^1,  yL  -i 

a"  "  b'*~ 
whenever  a'  —  b'.     In  other  words,  the  equation 


when  the  reference-axes  are  oblique,  denotes  an  ellipse 
referred  to  its  equi-  conjugate  diameters. 

42O.  By  throwing  the  equation  of  Art.  417  into  the 
form 


and  subjecting  it  to  an  analysis  precisely  like  that  of 
Art.  361,  we  can  determine  the  figure  of  the  Ellipse 
with  respect  to  any  two  conjugates.  It  will  thus  appear 
that  the  curve  is  oblong,  closed,  continuous  in  extent, 
and  symmetric  not  only  to  the  axis  major  and  axis  minor, 
but  to  any  diameter  whatever. 

CONJUGATE    PROPERTIES    OF    THE    TANGENT. 

421.  Equation  to  I  lie  Tangent,  referred  to  any 
two  Conjugate  Diameters.  —  From  the  relation  estab- 
lished (Art.  418)  between  the  equations 

-  =  1    and  =l 


340 


ANALYTIC  GEOMETRY. 


it  follows  that  the  application  to  the  latter  of  the  method 
used  in  Arts.  384,  385  must  result  in 

~ofi  +  W  =    l' 

422.  Intercept  of  the  Tangent  on  any  Diam- 
eter.— Making  y  =  0  in  the 
equation  just  found,  we  get, 
for  the  intercept  in  question, 


We  have,  then,  as  the  extension  of  Art.  391, 

Theorem  XLIII. — The  intercept  cut  off  ly  a  tangent 
upon  any  diameter  of  an  ellipse  is  a  third  proportional 
to  the  abscissa  of  contact  and  the  semi- diameter. 

Corollary. — To  construct  a  tangent  from  any  given  point. 
From  the  given  point  Tf,  draw  the  diameter  T'C,  and 
form  its  conjugate  CD'.  On  CTf  take  CM  a  third  pro- 
portional to  the  intercept  CT'  and  the  semi-diameter  CD, 
and  draw  MP  parallel  to  £ZX :  then  will  MP  be  the  ordi- 
nate  of  contact.  Join  T'P,  which  will  be  the  tangent 
required. 

423.  We  may  conveniently  group  at  this  point  a  few 
properties  of  tangents  and  their  intercepts,  which  will 
serve  to  illustrate  the  advantages  of  using  conjugate 
diameters  as  axes  of  reference. 

I.  Let  r/  T,  L'T'  be  any 
two  fixed  parallel  tangents  of 
an  ellipse,  intersected  by  any 
variable  tangent  T'T.  I/L' 
joining  the  points  of  contact 
of  the  parallel  tangents,  and 
DL  drawn  parallel  to  them, 


TANGENT  EEFEEEED  TO  CONJUGATES.      341 

(Art.  389)  will  be  conjugate  diameters.    Taking  these  for  reference- 
axes,  the  equation  to  the  variable  tangent  T'T  is 


In  this,  making  y  successively  equal  to  + 


—  V  ',  we  obtain 


Hence,  multiplying  together  the  two  values  of  x}  and  substituting 
for  y/2  its  value  from  the  equation  to  the  Ellipse,  we  get 


Interpreting  this  relation  in  ordinary  language,  we  have 

Theorem  XLIV, — The  rectangle  under  the  intercepts  cut  off  upon 
two  fixed  parallel  tangents  by  any  variable  tangent  of  an  ellipse  is 
constant,  and  equal  to  the  square  on  the  semi-diameter  parallel  to  the 
two  tangents. 

II.  If  we  take  for  axes  of  reference  the  diameter  CP  drawn  to 
the  point  of  contact  of  any  variable  tangent,  and  the  conjugate 
diameter  SS',  the  equations  to  any  two  fixed  parallel  tangents 
I/T,  L'T'  will  be 

^  4_  y'y  —  i    x'x  i  y'y  -  _  i 

an  T    £/,   ~:  1J        a,-i.  T   yi 

Making  x  =  a'  in  each  of  these,  and  remembering  (Art.  389)  that 
the  axis  of  y  (S/S')  is  parallel  to  the  variable  tangent  T'T,  we  get 


Hence,  after  substituting  for  t/2  from  the  equation  to  the  curve, 


the  sign  of  the  second  factor  being  disregarded,  as  we  are  only 
concerned  with  the  area  of  the  rectangle.     We  have,  then, 

Theorem  XLV  —  The  rectangle  under  the  intercepts  cut  off  upon 
any  variable  tangent  of  an  ellipse  by  two  fixed  parallel  tangents  is 
variable,  being  equal  to  the  square  on  the  semi-diameter  parallel  to  the 
tangent. 


342  ANALYTIC  GEOMETRY. 

III.  Using  the  same  axes  of  reference  as  in  II,  the  equations  to 
any  two  conjugate  diameters,  for  instance  CT  and  CT',  (Arts.  372, 
418)  may  be  written 

^_^_-0       ^+^.-0 
x/       y'~'    '       a"  T  6/2  " 

Making  x  =  of  in  each,  we  obtain 


Hence,  neglecting  the  sign  of  the  second  intercept, 

FT.  PT'  =  6/2, 
and  we  thus  arrive  at 

Theorem  XLVI,  —  The  rectangle  under  the  intercepts  cut  off  upon 
any  variable  tangent  of  an  ellipse  by  two  conjugate  diameters  is  equal 
to  the  square  on  the  semi-diameter  parallel  to  the  tangent. 

Remark.  —  It  is  evident  that,  by  a  single  change  in  the  interpre- 
tation of  the  symbols,  we  might  have  stated  the  theorem  thus: 
The  rectangle  under  the  intercepts  cut  off  upon  a  fixed  tangent  by  any 
two  conjugates  is  constant,  and  equal  to  the  square  on  the  parallel 
semi-diameter. 

Corollary  1.  —  It  is  obvious  from  the  equations,  that  none  but 
conjugate  diameters  will  cut  off  such  intercepts  as  will  form  the 
rectangle  mentioned.  Hence,  the  converse  of  the  theorem  is  also 
true;  and,  combining  it  with  the  result  of  II,  we  get:  Diameters 
drawn  through  the  intersections  of  any  tangent  with  two  parallel  tan- 
gents are  conjugate. 

Corollary  2.  —  The  theorem  of  III  also  furnishes  us  with  the  fol- 
lowing neat  solution  of  the  problem, 

Given  two  conjugate  diameters  of  an  ellipse  in  position  and  magni- 
tude, to  construct  the  axes.  Let  CD,  CD/  be  the  given  conjugate 
semi-diameters.  Through  the  extremity  of 
either,  as  D,  draw  AB  parallel  to  the  other: 
it  will  be  a  tangent  of  the  corresponding 
ellipse  (Art.  389).  Produce  CD  to  P,  so 
that  CD.  DP  =  CD'2.  Bisect  CP  by  the 
perpendicular  MO,  and  from  O,  where  MO 
meets  AB,  describe  a  circle  through  C  and 
P.  Join  the  points  A  and  B,  in  which  this 
circle  cuts  AB,  with  the  center  C  of  the  ellipse:  CA,  CB  will  be 


SUBTANGENT  IN  GENERAL. 


343 


the  axes  sought.  For  (Geom.,  331)  AD.DB  =  CD.DP=  OZX2,  and 
therefore  (Th.  XLVI)  CA,  CB  are  conjugate  diameters.  Moreover, 
since  ACB  is  inscribed  in  a  semicircle,  they  are  the  rectangular 
conjugates  of  the  curve;  or,  in  other  words,  the  axes. 

424.  Subtangent  to  any  Diameter. — This  being 
the   portion   of  any   diameter   intercepted   between   the 
foot  T'  of  a  tangent,  and  the  foot  M  of  the  ordinate 
of  contact,   we  have,  for   its 
length,   MT'  =  CTf  —  CM. 
Hence,  (Art.  422,) 


subtan'  = 


a'2  —  x'2 
~  x' 


That  is,  since  a1  +  x'  =  LM, 

and  a'  —  x'  =  MD,  The  subtangent  to  any  diameter  of  an 
ellipse  is  a  fourth  proportional  to  the  abscissa  of  contact 
and  the  corresponding  segments  of  the  diameter. 

Corollary. — This  value  is  independent  of  the  length 
of  the  conjugate  semi-diameter  6';  and,  if  we  compare 
it  with  that  of  the  subtangent  in  the  circle  x2  ~{-yz^=a'2 
described  upon  the  diameter  which  serves  as  the  axis 
of  x,  we  find  (Art.  311)  that  the  two  are  equal.  Hence 
the  following  construction : 

To  draiv  a  tangent  to  an  ellipse  from  any  given  point. 
Let  Tf  be  the  given  point.  Through  it  draw  the  diameter 
TfDL ;  upon  DL  describe  a  circle,  and  form  its  tangent 
TQ  passing  through  the  given  point.  Let  fall  QM  per- 
pendicular to  DL,  and  through  its  foot  M  draw  PP'  a 
double  ordinate  to  the  diameter  DL :  the  points  P,  P' 
in  which  this  meets  the  curve  will  be  the  points  of  con- 
tact of  the  two  possible  tangents  from  T',  either  of  which 
may  be  obtained  by  joining  T'  to  the  proper  point  of 
contact. 

An.  Go.  32. 


344  ANALYTIC  GEOMETRY. 

Remark,  —  It  is  obvious  that  the  same  principle  may 
be  used  to  construct  a  tangent  at  any  point  P  of  the 
curve,  by  simply  drawing  any  diameter  and  its  conjugate, 
forming  the  corresponding  ordinate  of  the  ellipse  from  _P, 
and  erecting  at  its  foot  the  ordinate  MQ  of  the  circle 
described  on  the  diameter  first  drawn  :  the  tangent  to 
this  circle  at  Q  will  determine  the  foot  Tf  of  the  required 
tangent  to  the  ellipse.  This  method  is  very  convenient 
when  the  curve  only,  or  an  arc  of  it,  is  given  ;  but,  when 
the  axes  are  known,  the  construction  described  in  the 
corollary  to  Art.  392  is  preferable. 

425.  If  we  multiply  the  value  of  the  subtangent  by 
the  abscissa  of  contact  a/,  we  get  xr  subtan'  =  a1'2  -  —  x'2. 
Comparing  this  with  the  square  on  the  ordinate  of 
contact,  as  given  by  the  equation  to  the  curve,  namely, 


we  obtain  x'  subtan'  :  y'2  =  a'2  :  &'2,  a  relation  expressed 

by 

Theorem  XL  VII.  —  -The  rectangle  under  the  subtangent 
and  the  abscissa  of  contact  is  fo  the  square  on  the  ordinate 
of  contact,  as  the  square  on  the  corresponding  diameter  is 
to  the  square  on  its  conjugate. 

426.  The  equations  to  the  tangents  at  the  extrem- 
ities of  any  chord  of  an  ellipse,  by  taking  for  axes  the 
diameter  parallel  to  the  chord,  and  its  conjugate,  may 
be  written 


b'2 


PARAMETER  OF  THE  ELLIPSE.  345 

Eliminating  between  these,  we  find  that  the  co-ordinates 
of  the  point  in  which  the  tangents  intersect  are 


Hence,  comparing  Arts.  369  ;  49,  Cor.  1,  we  have 

Theorem  XL  VIII.  —  Tangents  at  the  extremities  of  any 
chord  of  an  ellipse  meet  on  the  diameter  which  bisects  that 
chord. 

PARAMETERS. 

427.  Definitions.  —  The  Parameter  of  an  ellipse, 
with  respect  to  any  diameter,  is  a  third  proportional  to 
the  diameter  and  its  conjugate.  Thus,  if  a',  bf  denote 
the  lengths  of  any  two  conjugate  semi-diameters,  we 
shall  have,  for  the  value  of  the  corresponding  parameter, 


parameter  =  ^-     =  -^ 

The  parameter  with  respect  to  the  axis  major,  is  called 
the  principal  parameter  ;  or,  the  parameter  of  the  curve. 
We  shall  represent  its  length  by  the  symbol  4p. 

428.  From  the  definition  above,  we  have,  for  the  value 
of  the  parameter  of  the  Ellipse, 

4     =  ?*!. 
P~      a 

Hence,  (Art.  357,  Cor.,)  the  principal  parameter  is  iden- 
tical with  the  line  which  we  named  the  latus  rectum; 
that  is,  it  is  the  double  ordinate  drawn  through  the, 
focus  to  the  axis  major. 


346  ANALYTIC  GEOMETRY. 

429.  In  Art.  415,  we  proved  that  the  focal  chord 
(or  double  ordinate)  parallel  to  any  diameter  is  a  third 
proportional  to  the  axis  major  and  the  diameter.  Now 
(Art.  366)  the  axis  major  is  greater  than  any  other 
diameter  —  greater,  therefore,  than  the  diameter  conju- 
gate to  that  of  which  the  focal  chord  is  a  parallel, 
unless  the  chord  is  the  latus  rectum.  Hence, 

Theorem  XLIX. — No  parameter  of  an  ellipse,  except 
the  principal,  is  equal  in  value  to  the  corresponding 
focal  double  ordinate. 


POLE    AND    POLAR. 

430.  We  can  now  show  that  the  reciprocal  relation 
of  points  and  right  lines  which  we   established   (Arts. 
318  —  321)  in  the  case  of  the  Circle,  is  a  property  of 
the  generic  curve  of  which  the  Circle  is  only  a  particu- 
lar case/    We  shall  develop  the  conception  of  the  polar 
line  in  the  Ellipse  by  the  same  steps  as  in  the  former 
investigation. 

431.  Chord  of  Contact  in  the  Ellipse. — Let  x'yf 
be   the   fixed   point   from  which  the  two   tangents   that 
determine   the   chord   are   drawn,    and    x\y^   x2y2    their 
respective    points    of    contact.      Their   equations    (Art. 
421)  will  be 

x\*    ,    M  _  1       ?*?  _L  M  —  1 

a1'2         bf2~  a''2  "   b'2~ 

But  x'y'  being  upon  both  tangents,  we  have 

_-i       W*  ,    M'_-| 
—  ^          n  --       *  ~ 


POLAR  IN  THE  ELLIPSE.  347 

That  is,  the  co-ordinates  of  both  points  of  contact  satisfy 
the  equation 


This  is  therefore  the  equation  to  the  chord  of  contact. 

432.  Locus  of  the  Intersection  of  Tangents  to 
the  Ellipse.  —  Let  x'y'  be  the  fixed  point  through 
which  the  chord  of  contact  belonging  to  two  inter- 
secting tangents  is  drawn,  and  x\y^  the  intersection  of 
the  tangents.  The  equation  to  the  chord  (Art.  431) 
will  be 


and,  as  x'y'  is  on  the  chord,  we  shall  have  the  condition 

i^  +  ik^1' 

irrespective  of  the  direction  of  the  chord.  The  co-ordi- 
nates of  intersection  for  the  two  tangents  drawn  at  its 
extremities  must  therefore  always  satisfy  the  equation 

P  +  fl  =  1: 

which  is  for  that  reason  the  equation  to  the  required 
locus. 

433.  Tangent  and  Chord  of  Contact  included 
in  the  wider  conception  of  the  Polar. — The  equa- 
tions to  the  tangent,  to  the  chord  of  contact,  and  to  the 
locus  of  the  intersection  of  tangents  drawn  at  the  ex- 
tremities of  chords  that  pass  through  a  fixed  point,  are 
thus  seen  to  be  identical  in  form.  These  three  lines  are 
therefore  only  different  expressions  of  a  common  formal 
law ;  and,  inasmuch  as  the  fixed  point  x'y' ,  in  the  case 


348  ANALYTIC  GEOMETRY. 

of  the  chord  of  contact,  is  restricted  to  being  without 
the  curve ;  and,  in  that  of  the  tangent,  to  being  on  the 
curve ;  while,  in  the  case  of  the  locus  in  question,  it  is 
not  restricted  at  all:  it  follows  that  the  tangent  and 
chord  of  contact  are  cases  of  the  locus,  due  to  bringing 
the  point  x'y'  upon  the  curve  or  outside  of  it,  More- 
over, the  relation  between  the  locus  which  thus  absorbs 
the  tangent  and  chord  of  contact,  and  the  fixed  point 
x'y',  is  that  of  polar  reciprocity.  For,  by  precisely  the 
same  argument  as  that  used  (Art.  321)  in  the  case  of 
the  Circle,  we  have  the  twofold  theorem : 

I.  If  from  a  fixed  point  chords  be  drawn  to  any  ellipse, 
and  tangents  to  the  curve  be  formed  at  the  extremities  of  each 
chord,  the  intersections  of  the  several  pairs  of  tangents  will 
lie  on  one  right  line. 

II.  If  from  different  points  lying  on  one  right  line  pairs 
of  tangents  be  drawn  to  any  ellipse,  their  several  chords 
of  contact  will  meet  in  one  point. 

It  thus  appears  that  the  Ellipse  imparts  to  every  point 
in  its  plane  the  power  of  determining  a  right  line ;  and 
reciprocally. 

434.  Equation  to  the  Polar  with  respect  to  an 
Ellipse. — From  what  has  been  shown  in  the  preceding 
articles,  it  is  evident  that  this  equation,  referred  to  any 
pair  of  conjugate  diameters,  is 


x'y'  being  the  point  to  which  the  polar  corresponds. 
Consequently,  the  equation  referred  to  the  axes  of  the 
curve  will  be 

x'x       y'y  _ 


POLAR  IN  THE  ELLIPSE. 


349 


435.  Definitions. — The  Polar  of  any  point,  with 
respect  to  an  ellipse,  is  the  right  line  which  forms 
the  locus  of  the  intersection  of  the  two  tangents 
drawn  at  the  extremities  of  any  chord  passing  through 
the  point. 

The  Pole  of  any  right  line,  with  respect  to  an  ellipse, 
is  the  point  in  which  all  the  chords  of  contact  corre- 
sponding to  different  points  on  the  line  intersect  each 
other. 

From  these  definitions,  we 
obtain  the  following  con- 
structions :  — When  the  pole 
P  is .  given,  draw  through  it 
any  two  chords  T'T,  S'S, 
and  form  the  corresponding 
pairs  of  tangents,  T'L  and 
TL,  S'M  and  SM.  Join  the 
intersection  of  the  first  pair 

to  that  of  the  second,  forming  the  line  LM :  this  will  be 
the  polar  of  P.  When  the  polar  is  given,  take  any  two 
points  upon  it,  as  L  and  M,  and  from  each  draw  a  pair 
of  tangents  to  the  curve  :  the  point  P,  in  which  the  cor- 
responding chords  of  contact  T'T,  S'S  intersect,  will  be 
the  pole  of  LM. 

In  case  the  pole  is  without  the  curve,  as  at  L,  the  polar 
is  the  chord  of  contact  of  the  two  tangents  from  L ;  and, 
when  the  pole  is  on  the  curve,  as  at  T,  the  polar  is  the 
tangent  at  T.  In  either  case,  then,  the  construction  may 
be  made  in  the  way  these  facts  require. 

436.  Direction  of  the  Polar. — The  equation  to 
the  polar  of  any  point  x'y',  namely  (Art.  434), 

x'x        y'y 


350  ANALYTIC  GEOMETRY. 

when  compared  with  that  of  the  diameter  conjugate  to 
the  same  point,  namely  (Arts.  372,  418), 

x'x        tfy  _ 

a'*  "     V* 

shows  (Art.  98,  Cor.)  that  the  polar  and  the  diameter 
are  parallel.  We  have,  then,  the  following  extension 
of  the  property  reached  in  Art.  389  : 

Theorem  L.  —  The  polar  of  any  point,  with  respect  to 
an  ellipse,  is  parallel  to  the  diameter  conjugate  to  that 
which  passes  through  the  point. 

437.  Polars  of  Special  Points.  —  It  is  easy  to  see, 
by  comparing  the  equations  to  the  polar  in  the  Ellipse 
and  in  the  Circle  (Arts.  434,  323),  that  the  general 
properties  of  polars  proved  in  Art.  324  are  true  in  the 
case  of  the  Ellipse.  We  leave  the  student  to  convince 
himself  of  this,  and  will  here  present  certain  special 
properties  of  polars,  which  depend  on  taking  the  pole 
at  particular  points. 

If  we  substitute  for  x'y',  in  the  equation  of  Art.  434, 
the  co-ordinates  of  the  center,  we  shall  get  1  =  0:  an 
expression  conforming  to  the  type  (Art.  110) 

0=0. 

Hence,  The  polar  of  the  center  is  a  right  line  at  infinity. 
If  in  the  same  equation  we  make  y'=  0,  we  shall  get 


Hence,  The  polar  of  any  point  on  a  diameter  is  a  right 
line  parallel  to  the  conjugate  diameter,  and  its  distance  from 
the  center  is  a  third  proportional  to  the  distance  of  the  point 
and  the  length  of  the  semi-diameter. 


POLAR  OF  THE  FOCUS. 


351 


Similary,  The  polar  of  any  point  on  the  axis  major  is 
the  perpendicular  whose  distance  from  the  center  is  a  third 
proportional  to  the  distance  of  the  point  and  the  length 
of  the  semi-axis. 

Corollary, — The  second  of  these  properties  obviously 
leads  to  the  following  construction  of  the  polar: — Join 
the  given  point  with  the  center  of  the  curve,  and,  from 
the  latter  as  origin,  lay  off  upon  the  resulting  diameter 
a  third  proportional  to  the  distance  of  the  point  and  the 
length  of  the  semi-diameter.  Through  the  point  thus 
reached,  draw  a  parallel  to  the  conjugate  diameter, 
which  will  be  the  polar  required. 

4H8.  Polar  of  tlie  Focus. — The  equation  to  the 
polar  of  either  focus,  by  substituting  (±  ae,  0)  for  x'y1 
in  the  second  equation  of  Art.  434,  is  found  to  be 


Hence,  The  polar  of  either  focus  in  an  ellipse  is  the 
perpendicular  ivhich  cuts  the  axis  major  at  a  distance 
from  the  center  equal  to  a  :  e,  measured  on  the  same 
side  as  the  focus. 

439.  The  distance  of  any  point  P  of  the  curve  from 
either  focal  polar,  say  DR, 
is  evidently  equal  to  the 
distance  of  that  polar  from 
the  center,  diminished  by  the 
abscissa  of  the  point.  Thus, 


ex 


Now  (Art.  360)  a  —  ex  =  FP.     Therefore, 
FP  _ 

PD~  e' 

An.  Ge.  33. 


352 


ANALYTIC  GEOMETRY. 


and  we  have  the  remarkable  property,  which  will  here- 
after be  found  to  characterize  all  the  Conies, 

Theorem  LI. —  The  distance  of  any  point  on  an  ellipse 
from  the  focus  is  in  a  constant  ratio  to  its  distance  from 
the  polar  of  the  focus,  the  ratio  being  equal  to  the  eccen- 
tricity of  the  curve. 

Corollary  1. — Upon  this  theorem  is  founded  the  follow- 
ing method  for  constructing  any  arc  of  an  ellipse.  The 
process  is  not  simple  enough  for  extensive  use,  but  is 
interesting  as  exhibiting  the  analogy  between  the  Ellipse 
and  the  other  two  Conies  in  regard  to  the  important 
property  just  established. 

Take  any  point  F,  and  any  fixed 
right  line  DR.  Draw  FR  perpendic- 
ular to  DR,  and,  at  any  convenient 
point  of  the  latter,  as  D,  make  DP 
parallel  to  FR  and  greater  than  FP, 
to  express  the  property  that  the  eccen- 
tricity of  the  Ellipse  (which,  by  the 
theorem  above,  equals  FP:  PD)  is  less 
than  unity.  On  PD  describe  a  semi- 
circle, and  from  P  as  center,  with  a 

radius  FP,  form  an  arc  cutting  the  semicircle  in  0.  Join  DO,  PO, 
producing  the  latter  to  meet  DR  in  M:  the  triangle  DOP  will  be 
right-angled  at  O,  being  inscribed  in  a  semicircle.  Now  divide  FR 
in  the  ratio  FP:  PD,  suppose  at  A:  then  will  A  be  the  vertex  of 
the  ellipse  of  which  P  is  a  point,  F  the  focus,  and  DR  the  polar 
of  the  focus.  At  any  point  on  the  line  RF  to  the  right  of  A,  as 
at  B,  erect  a  perpendicular,  meeting  PM  in  C.  Draw  CE  parallel 
to  PD,  and  EG  parallel  to  DO.  Next,  from  F  as  center,  with  a 
radius  equal  to  CG,  describe  an  arc  cutting  BC  in  the  point  a: 
then  will  a  be  a  point  on  the  curve.  For  the  triangle  CEG  is  by 
construction  similar  to  PDO,  so  that  CG  :  CE  =  PO  :  PD;  or, 
Fa  :  CE  =  FP:  PD.  Thus  the  focal  distance  of  the  point  is  to  its 
distance  from  the  line  which  may  now  be  considered  the  focal  polar, 
in  a  constant  ratio  less  than  unity.  By  repeating  the  process  just 
described,  other  points  of  the  curve  may  be  found  in  sufficient 


ELLIPSE  DEFINED  BY  POLE  AND  POLAR.     353 

numbers   to  determine   its   outline,   and   it   can   then    be    drawn 
through  them. 

Corollary  2. — The  fact  has  been  brought  to  light 
(Arts.  431  —  433),  that  the  positions  of  the  pole  and 
polar  may  be  as  near  to  or  as  remote  from  each  other 
as  we  please.  It  is  therefore  not  only  true  that  an 
ellipse  imparts  to  every  point  in  its  plane  the  power 
of  determining  a  right  line,  but  any  given  right  line 
is  the  polar  of  any  given  point,  with  respect  to  some 
ellipse.  Now  the  construction  just  explained  rests  upon 
this  principle,  and  the  given  line  DE  is  therefore  con- 
stantly taken  as  the  boundary  against  which,  as  the 
polar  of  the  given  point  F,  the  horizontal  distances 
of  the  points  on  the  curve  are  measured.  From  this 
constant  relation  to  the  figure  of  the  resulting  ellipse, 
the  polar  of  the  focus  is  called  the  directrix  of  the 
curve. 

Corollary  3. — The  theorem  of  this  article  invests  the 
term  ellipse  with  a  new  meaning.  We  now  see  that  the 
name  of  the  curve  may  be  interpreted  as  signifying  the 
conic  in  which  the  constant  ratio  between  the  focal  and 
polar  distances  falls  short  of  unity. 

44O.    Focal  Angle   subtended  by  any  Tangent. — By  this 
is  meant  the  angle  PFT  included  between  two  focal  radii  FT, 
FP:  one  drawn  to  the  point  of  contact 
of  the    tangent   passing    through  any 
fixed  point  P;  and  the  other,  to  such 
fixed  point  itself.     The  determination 
of  this    angle   involves    a    remarkable 
relation,  which,  although  not  depend- 
ing upon  the  properties  of  conjugate 
diameters,  we  shall  nevertheless  present 

here,  because  the  proper  statement  of  it  implies  that  the  reader  is 
acquainted  with  the  definition  of  the  pole  and  the  polar. 


354 


ANALYTIC  GEOMETRY. 


Let  xy  be  the  arbitrary  point  P  through  which  a  tangent  PT  is 
drawn,  and  x'y'  the  corresponding  point  of  contact  T.     Also,  let 


We  shall  then  have 


=  FP,   and    p'  = 


008*  =  *       -*,      sinfl=^; 
P  P 

ae — a/ 


Therefore,  putting  0  =  P^T7,  we  shall 
get  (Trig.,  845,  iv) 

cos  ^  =  (ae  ~  g)  (ae 


Now,  from  the  equation  to  the  tangent,  y'y  —  I*  --  r23^x.     Hence, 

pp'  cos  $  —  a2  —  aea:  —  ae  a/  +  e2a/a;  =  (a  — 
But  (Art.  360)  /  =  a  —  ea/.     Hence,  finally, 


441.  This  expression,  being  independent  of  the  point  of  contact 
a/y',  must  be  true  for  either  of  the  tangents  drawn  from  P.  Hence 
?  =PFT=PFT/.  Therefore,  with  respect  to  the  whole  angle  TFT', 
we  have 

Theorem  LII.  —  The  right  line  that  joins  the  focus  of  an  ellipse  to 
the  pole  of  any  chord,  bisects  the  focal  anyle  which  the  chord  subtends. 

Corollary. — Since  the  angle  subtended  by  any  focal  chord  is  180°, 
we  obtain  the  further  special  property :  The  line  that  joins  the  focus 
to  the  pole  of  any  focal  chord  is  perpendicular  to  the  chord. 


in.  THE  CURVE  REFERRED  TO  ITS  Foci. 

443.  We  are  now  prepared  to  attach  the  proper 
meanings  to  the  constants  which  enter  the  equations  of 
Art.  152  and  the  subjoined  Remark;  and  it  may  deserve 
to  be  mentioned  in  passing,  that  the  phrases  polar  equa- 
tion, polar  co-ordinates,  etc.,  have  no  reference  to  the  polar 
relation  lately  developed  as  a  property  of  the  Ellipse. 


POLAR  EQUATION  TO  TANGENT. 


355 


443.  From  the  preceding  discussions,  then,  we  are 
henceforth  to  understand  that  in  the  polar  equations 

a  (I  —  e2)  a  (I—  e2) 

Q      __  >•  ___  '_  f)      -  V  _  / 

1  —  ecos#  1-f-ecos^' 

the  constant  a  is  the  semi-axis  major  of  the  given  ellipse, 
and  the  constant  e  its  eccentricity. 

Further  :  If  we  replace  1  —  er  by  its  value  (Art.  151) 
62  :  #2,  we  may  write  these  equations 


?  =  a 


b2 


1  —  ecostf  a     1-j-ccos^ 

Now  (Art.  428)  b2  :  a  is  half  the  parameter  of  the  curve. 


±  e  cos 


the  upper  or  lower  sign  being  used  according  as  the 
right-hand  or  left-hand  focus  is  taken  for  the  pole. 

444.  Polar  Equation  to  the  Tangent.  —  We  shall 
obtain  this  most  readily  by  transforming  the  equation 


from  rectangular  to  polar  co-ordinates,  at  the  same  time 
removing  the  pole  to  the  left-hand  focus,  whose  co-ordi- 
nates are  —  ae,  0.  We  have  (Art.  58)  a/  =  //cos  6'  —  ae* 
x  =  p  cos  6  —  ae,  yf  =  pr  sin  6f,  y  =  p  sin  6.  Hence,  the 
transformed  equation  is 


(f*f  cos  6'  —  ae)  (p  cos  0 


, 


p'p  COS  d  COS  6* 


1. 


But  p'O'  is  on  the  curve  :  therefore,  (Art.  443,) 


—  e 


356 


ANALYTIC  GEOMETRY. 


Substituting  these  values  in  the  first  and  second  terms 
of  the  last  equation  respectively,  and  reducing, 

(cosdf — e)  (f)  cosd — ae)-\-p  sin#  sin  d'=a  (1 — ecos#r)  (1). 
Therefore,  (Trig.,  845,  iv,)  the  required  equation  is 

_  a  (I-*2) 

cos  (6  —6'}—e  cos  6  ' 

Corollary, — Since  the  equation  to  the  diameter  conju- 
gate to  x'y'  (Art.  372),  differs  from  that  of  the  tangent 
at  x'y'  only  in  having  0  for  its  constant  term,  we  have, 
by  putting  0  instead  of  a  (1 — e  cos  6')  in  (1),  and  reducing, 

ae  (cos  6'  —  e) 
P  ~~=  cos  (0  —  6'}—e  cos  6  ' 

as  the  polar  equation  to   the,  diameter  conjugate  to  that 
which  passes  through  p'6'. 


These  polar  equations  afford  a  proof  of  Theorem  XIX 
so  much  simpler  than  the  one  given  in  Art.  378,  that  we  present 
it  here  expressly  to  invite  comparison. 

Let  p/0/  be  the  extremity  D  of  any 
diameter  DIY ,  The  equation  to  its 
conjugate  SS'  is 

ae  (cos  0*  —  e) 


cos  (#  —  6/)  —  e  cos  0 
In  this,  making  d  —  d*,  we  obtain 


S' 


Now,  from  the  polar  equation  to  the  curve,  as  given  in  Art.  152, 


1  —  e  cos  (? 


Hence, 


AREA  OF  THE  ELLIPSE. 


357 


Remark. — The  geometric  proof  of  this  theorem  is  perhaps  still 
simpler.  Thus: — The  normal  DN  bisects  the  angle  F'DF  (Art. 
403),  and  is  perpendicular  to  SS'  (Art.  389).  Hence,  the  triangle 
MDL  is  isosceles,  and  I)M=DL.  But,  by  drawing  a  parallel  to  LM 
through  F,  it  becomes  apparent,  since  CF/=CF^  that  F'M—  FL. 
Therefore,  DM  +  DL  =  F'D  —  F'M  +  FD  +  FL  =F'D  +  FD. 
That  is,  (Art.  355), 

2DM=A'A  .-.  DM=a. 

We  have  given  these  three  proofs  of  the  same  proposition  for  the 
purpose  of  illustrating  the  importance  of  a  proper  selection  of  methods, 
even  in  the  elementary  work  of  the  beginner.  In  attempting  to  es- 
tablish theorems,  several  methods  are  often  available  to  the  student, 
and  he  should  select  that  which  will  combine  rigor,  simplicity,  and 
elegance,  in  the  highest  degree.  While  analytic  processes  are  gen- 
erally able  to  satisfy  this  condition  the  best,  it  nevertheless  sometimes 
happens,  that  the  proof  from  pure  geometry  is  superior.  In  such  a 
case,  of  course,  the  latter  is  to  be  preferred. 


A' 


M 


iv.    AREA  OF  THE  ELLIPSE. 

446.  The  area  of  any  ellipse  whose  axes  are  given, 
may  be  determined  by  the  following  application  of  the 
geometric  method  of  infinitesimals. 

A  A  being  the  axis  major  of  the 
curve,  describe  a  circle  upon  it,  and 
divide  it  into  any  number  of  equal 
parts  at  Z,  M,  N,  etc.  Erect  the 
ordinates  ZP,  M Q,  NR,  etc.,  cutting  the  ellipse  in  p,  q,  r, 
etc.  Join  PQ,  pq :  then,  since  Lp  :  LP=  Mq  :  MQ  =  b  :  a 
(Art.  359),  the  trapezoids  Lq,  LQ  are  in  the  ratio  b :  a. 
Now  the  same  must  be  true  of  any  two  corresponding 
trapezoids:  therefore,  the  area  of  the  polygon  inscribed 
in  the  ellipse  is  to  the  area  of  the  corresponding  polygon 
inscribed  in  the  circle  as  b  is  to  a.  Hence,  as  this  pro- 
portion holds  true,  no  matter  how  many  sides  the  inscribed 


358  ANALYTIC   GEOMETRY. 

polygons  may  have,  and  as  we  can  make  the  number  of 
sides  as  great  as  we  please  by  continually  subdividing 
the  axis  major,  in  the  limiting  case  where  the  polygons 
vanish  into  their  respective  curves,  we  shall  have  area 
of  ellipse  :  area  of  circumscribed  circle  =  b  :  a.  Now 
(Geom.,  500)  the  area  of  the  circle  —  no,2.  Therefore, 
putting  A  =  the  required  area, 


That  is, 

Theorem  LIII.  —  The  area  of  an  ellipse  is  equal  to  n  times 
the  rectangle  under  Us  semi-axes. 


Corollary. — Since  rtab  =  i/7ra2.  ;r62,  we  have  the  addi- 
tional property  :  The  area  of  an  ellipse  is  a  geometric 
mean  between  the  areas  of  its  circumscribed  and  inscribed 
circles. 

EXAMPLES     ON     THE     ELLIPSE. 

1.  Find  the  equations  to  the  tangent  and  normal  at  the  extremity 
of  the  latus  rectum,  and  determine  the  eccentricity  of  the  ellipse  in 
which -the  normal  mentioned  passes  through  the  extremity  of  the 
axis  minor. 

2.  Find  the  equations  to  the  diameter  passing  through  the  ex- 
tremity of  the  latus  rectum,  and  the  chord  joining  the  extremities 
of  the  axes ;  and  determine  the  eccentricity  of  the  ellipse  in  which 
these  lines  are  parallel. 

3.  A  point  P  is  so  taken  on  the  normal  of  an  ellipse,  that  its 
distance  from  the  foot  of  the  normal  is  in  a  constant  ratio  to  the 
length  of  the  normal :  find  the  locus  of  P,  and  prove  that  when  P 
is  the  middle  point  of  the  normal,  its  locus  is  an  ellipse  whose  eccen- 
tricity e'  is  connected  with  that  of  the  given  one  by  the  condition 

(1—  <5/2)(l+e2)2=l  —  e\ 

4.  Prove  that  two  ellipses  of  equal  eccentricity  and  parallel  axes 
can  have  only  two  points  in  common.    Also,  show  that  if  three  such 
ellipses  intersect,  their  three  common  chords  will  meet  in  one  point. 


EXAMPLES  ON  THE  ELLIPSE. 


359 


5.  If  two  parallels  bo  drawn,  one  from  an  extremity  A  of  the 
axis  major,  and  the  other  from  the  adjacent  focus,  meeting  the  axis 
minor  in  M  and  JV,  the  circle  described  from  N  as  a  center,  with  a 
radius  equal  to  MA,  will  either  touch  the  ellipse  or  fall  entirely 
outside  of  it. 

6.  A  circle  is  inscribed  in  the  triangle  formed  by  the  axis  major 
and  any  two  focal  radii  :  find  the  locus  of  its  center. 

7.  Find  a  point  on  an  ellipse,  such  that  the  tangent  there  may 
be  equally  inclined  to  both  axes.    Also,  a  point  such  that  the  tangent 
may  form  upon  the  axes  intercepts  proportional  to  them. 

8.  Prove  that  the  circle  described  on  any  focal  radius  of  an  ellipse 
touches  the  circumscribed  circle. 

9.  From  the  vertex  of  an  ellipse  a  chord  is  drawn  to  any  point 
on  the  curve,  and  the  parallel  diameter  is  also  drawn:  the  locus 
of  the  intersection  of  this  diameter  and  the  tangent  at  the  extrem- 
ity of  the  chord,  is  the  tangent  at  the  opposite  vertex. 

10.  From  the  center  of  an  ellipse,  two  radii  vectores  are  drawn 
at  right  angles  to  each  other,  and  tangents  to  the  curve  are  formed 
at  their  extremities  :  the  tangents  intersect  on  the  ellipse 


11.  Two  ellipses  have  a  common  center,  and  axes  coincident  in 
direction,  while  the  sum  of  the  squares  on  the  axes  is  the  same  in 
both:  find  the  equation  to  a  common  tangent. 

12.  The  ordinate  of  any  point  P  on  an  ellipse  is  produced  to 
meet  the  circumscribed  circle  in  Q  :  the  focal  perpendicular  upon 
the  tangent  at  Q  is  equal  to  the  focal  distance  of  P. 

13.  The  lines  which  join  transversely  the  foci  and  the  feet  of  the 
focal  perpendiculars  on  any  tangent,  intersect  on  the  corresponding 
normal,  and  bisect  it. 

14.  If  a  right  line  drawn  from  the  focus  of  an  ellipse  meets  the 
tangent  at  a  constant  angle  0,  the  locus  of  its  foot  is  a  circle,  which 
touches  the  curve  or  falls  entirely  outside  of  it  according  as  cos  6  is 
less  or  greater  than  e. 

15.  When  the  angle  between  a  tangent  and  its  focal  radius  of 
contact  is  least,  the  radius  =  a;    and  when  the  angle  between  a 
tangent  and  its  central  radius  of  contact  is  least,  the   radius  = 
J  V  '  dl  -f-  P. 


360  ANALYTIC  GEOMETRY. 

16.  The  locus  of  the  foot  of  the  central  perpendicular  upon  any 
tangent  to  an  ellipse,  is  the  curve 


17.  The  locus  of  the  variable  intersection  of  two  circles  described 
on  two  conjugate  semi-diameters  of  an  ellipse,  is  the  curve 


18.  If  lines  drawn  through  any  point  of  an  ellipse  to  the  extrem- 
ities of  any  diameter  meet  the  conjugate  CD  in  M  and  N,  prove 
that  CM.CN  =  CD-. 

19.  In  an  ellipse,  the  rectangle  under  the  central  perpendicular 
upon  any  tangent  and  the  part  of  the  corresponding  normal  inter- 
cepted between  the  axes,  is  constant,  and  equal  to  a2  —  I'*. 

20.  The  condition  that  two  diameters  of  an  ellipse  may  be  conju- 
gate, referred  to  a  pair  of  conjugates  as  axes  of  co-ordinates,  is 

bn 
tan  6  tan  &  =  --  ^  • 

21.  Normals  at  P  and  D,  the  extremities  of  conjugate  diameters, 
meet  in  Q:  prove  that  the  diameter  CQ  is  perpendicular  to  PD, 
and  find  the  locus  of  its  intersection  with  the  latter. 

22.  Given  any  two  semi-diameters,  if  from  the  extremity  of  each 
an  ordinate  be  drawn  to  the  other,  the  triangles  so  formed  will  be 
equal  in  area.     Also,  if  tangents  be  drawn  at  the  extremity  of  each, 
the  triangles  so  formed  will  be  equal  in  area. 

23.  Find  the  locus  of  the  intersection  of  the  focal  perpendicular 
upon  any  tangent  with  the  radius  vector  from  the  center  to  the  point 
of  contact.    Also,  the  locus  of  the  intersection  of  the  central  perpen- 
dicular with  the  radius  vector  from  the  focus  to  the  point  of  contact. 

24.  The  equi-conjugates  being  taken  for  axes,  find  the  equation 
to  the  normal  at  P,  and  prove  that  the   normal  bisects  the  line 
joining  the  feet  of  the  perpendiculars  dropped  from  P  upon  the 
equi-conjugates. 

25.  Find  the  locus  of  the  intersection  of  tangents  drawn  through 
the  extremities  of  conjugate  diameters. 

26.  Putting  p,  p'  to  denote  the  focal  radii  of  any  point  on  an 
ellipse,  and  <j>  for  its  eccentric  angle,  prove  that 


EXAMPLES  ON  THE  ELLIPSE.  361 

27.  Express  the  lengths  of  two  conjugate  semi-diameters  in  terms 
of  the  eccentric  angle,  namely,  by 

a/2  =  a~  cos2  9  +  I2  sin2  0,       b/2  =  a2  sin2  0  -\-  Ir  cos2  <^>. 

28.  The  ordinate  MP  of  an  ellipse  being  produced  to  meet  the 
circumscribed  circle  in  Q,  find  the  locus  of  the  intersection  of  the 
radius  CQ  with  the  focal  radius  FP. 

29.  Normals  to  the  ellipse  and  circumscribed  circle  pass  through 
the  points  P  and  Q  just  mentioned,  and  intersect  in  R:   find  the 
locus  of  E. 

[First  show  that  the  equation  to  the  normal  of  the  ellipse  is 

_f?L_Jy_  =  c,, 

cos  (j)      sin  (f> 
0  being  the  eccentric  angle  of  the  point  of  contact.] 

30.  Prove  that   the   area  of  any  parallelogram  circumscribed 
about  an  ellipse  may  be  expressed  by 

4ab 


area  = 


where  0,  tf  are  the  eccentric  angles  corresponding  to  the  points  of 
contact  of  the  adjacent  sides.  Show  that  this  area  is  least  when 
the  points  of  contact  are  the  extremities  of  conjugates. 

31.  Upon  the  axis  major  of  an  ellipse,  two  supplemental  chords 
are  erected,  and  perpendiculars  are  drawn  to  them  from  the  vertices : 
show  that  the  locus  of  the  intersection  of  these  perpendiculars  is 
another  ellipse,  and  find  its  axes. 

32.  Let  CP,    CD   be    any  two    conjugate   semi-diameters :    the 
supplemental  chords  from  P  to  the  extremities  of  any  diameter  are 
parallel  to  those  from  D  to  the  extremities  of  the  conjugate. 

33.  The  rectangle  under  the  segments  of  any  focal  chord  is  to 
the  whole  chord  in  a  constant  ratio. 

34.  The  sum  of  two  focal  chords  drawn  parallel  to  two  conjugate 
diameters  is  constant. 

35.  The  sum  of  the  reciprocals  of  two  focal  chords  at  right  angles 
to  each  other  is  constant. 

36.  To  a  series  of  confocal  ellipses,  tangents  are  drawn  from  a 
fixed  point  on  the  axis  major :  find  the  locus  of  the  points  of  contact. 


362  ANALYTIC  GEOMETRY. 

37.  Tangents  to  two  confocal  ellipses  are  drawn  to  cut  each  other 
at  right  angles :  the  locus  of  their  intersection  is  a  circle  concentric 
with  the  ellipses. 

38.  Find   the  sum  of  the  focal  perpendiculars  upon  the  polar 
of  x't/. 

39.  The  intercept  formed  on  any  variable  tangent  by  two  fixed 
tangents,  subtends  a  constant  angle  at  the  focus.     Also,  the  line 
which  joins  the  focus  to  the  point  in  which  any  chord  cuts  the 
directrix,  is  the  external  bisector  of  the  focal  angle  subtended  by 
the  chord. 

40.  One  vertex  of  a  circumscribed   parallelogram   moves  along 
one  directrix  of  an  ellipse :    prove  that  the  opposite  vertex  moves 
along  the  other,  and  that  the  two  remaining  vertices  move  upon  the 
circumscribed  circle. 


CHAPTER   FOURTH. 

THE   HYPERBOLA, 
i.  THE  CURVE  REFERRED  TO  ITS  AXES. 

447.  In  discussing  the  Hyperbola  by  means  of  its 
equation  (Art.  167) 

f!        2!  —  1 
a2        b* 

we  shall  avoid  the  repetition  of  much  that  has  already 
been  said  in  connection  with  the  Ellipse,  by  considering 
that  the  similarity  of  the  equations  to  these  two  curves 
makes  most  of  the  arguments  used  in  the  foregoing  pages 
at  once  applicable  to  the  Hyperbola.  We  shall  therefore 
avail  ourselves  of  the  principle  developed  in  the  corollary 
to  Art.  167,  and,  for  details,  shall  refer  the  student  to 
the  proper  article  in  the  preceding  Chapter. 

For  the  sake  of  bringing  out  the  antithesis  between 
the  Ellipse  and  Hyperbola,  alluded  to   in  the  Remark 


PROPERTIES  OF  THE  HYPERBOLA.  363 

under  Art.  167,  the  theorems  of  this  Chapter  are  num- 
bered like  the  corresponding  ones  of  the  preceding. 

THE    AXES. 

448.  Making  y  and  x  successively  equal  to  zero  in 
the  equation  of  Art.  167,  we  get,  for  the  intercepts  of 
the  Hyperbola  upon  the  lines  termed  its  axes, 

x=~-±:a,     y  =--  ±  b  l/::rL 

Hence,  the  curve  cuts  the  transverse  axis  in  two  rea 
points  equally  distant  from  the  focal  center,  and  the 
conjugate  axis  in  twro  imaginary  points  situated  on 
opposite  sides  of  that  center  at  the  distance  bi/--I. 
Assuming,  then,  the  conjugate  axis  to  be  measured  by 
the  imaginary  unit  1/--1,  we  may  infer 

Theorem  I. — The  focal  center  of  any  hyperbola  bisects 
the  transverse  axis,  and  also  the  conjugate. 

Corollary. — In  the  light  of  the  analysis  leading  to  this 
theorem,  we  should  therefore  interpret  the  constants  a 
and  b  in  the  equation 

*2        f_, 
a2  "  b2  ~ 

as  respectively  denoting  half  the  transverse  axis  and 
half  the  modulus  of  the  imaginary  conjugate  axis. 

449.  At  the   outset   (see    Art.   166),   we   arbitrarily 
used  the  phrase  conjugate  axis  to 

denote  the  whole  line  drawn  through 
the  center  C  at  right  angles  to  the 
transverse  axis  A' A.  We  now  see 
that  the  phrase  in  strictness  means 
an  imaginary  portion  of  that  line, 
of  the  length  =  2&T/:=:1. 

But,  as  was  promised  in  Art.  166,  we  shall  now  show 


364  ANALYTIC  GEOMETRY. 

that  a  certain  real  portion  of  this  line  has  a  most  signifi- 
cant relation  to  the  Hyperbola,  on  account  of  which  it  is 
by  universal  consent  taken  for  the  conjugate  axis.  This 
relation  depends  on  the  companion-curve  called  the  conju- 
gate hyperbola,  whose  equation  we  developed  in  Art.  168. 
By  referring  to  the  close  of  Art.  168,  it  will  be  seen 
that  the  equations  to  two  conjugate  hyperbolas,  when 
referred  to  their  common  center  and  axes,  differ  only  in 
the  sign  of  the  constant  term.  The  equation  to  the  curve 
whose  branches  lie  one  above  and  the  other  below  the 
line  A1  A  in  the  diagram,  may  therefore  be  written 

—    -y--    _1 
a"  ~     b2~ 

Now,  if  in  this  we  make  x  =  0,  we  get 


Hence,  the  conjugate  hyperbola  has  a  real  axis,  identical 
in  direction  with  the  imaginary  axis  of  the  primary  curve, 
whose  length  is  the  same  multiple  of  1  that  the  length 
of  the  imaginary  is  of  V  —  1.  Moreover,  it  is  found  that 
this  real  axis  of  the  conjugate  hyperbola,  when  used  in- 
stead of  the  imaginary  one  of  the  primary  curve,  enables 
us  to  state  the  properties  of  the  latter  in  complete  analogy 
to  those  of  the  Ellipse.  It  is  customary,  therefore,  to  lay 
off  CB,  CB1  each  equal  to  6,  and  to  treat  the  resulting 
line  B'B  as  the  conjugate  axis  of  the  original  hyperbola, 
though  in  fact  it  is  only  the  transverse  axis  of  the  conju- 
gate curve. 

Adopting  this  convention,  the  statement  in  Theorem  I 
is  to  be  taken  without  reference  to  imaginary  quantities, 
and  the  constants  a  and  b  in  the  equations 


b2 


CONSTRUCTION  OF  CURVE  AND  FOCI.          365 

are  henceforth  to  be  interpreted  as  denoting  the  semi-axes 
of  the  curve. 

45O.  If  in  the  equation  of  Art.  167,  which  may  be 
itten 


written 

b 


—  a2, 


we  suppose  x  <C  a  or  >  —  #,  the  corresponding  values 
of  y  are  imaginary  ;  so  that  no  point  of  the  curve  is 
nearer  to  the  origin,  either  on  the  right  or  on  the  left, 
than  the  extremities  of  the  transverse  axis.  But  (Art. 
171),  for  the  distance  from  the  origin  to  either  focus, 
we  have 

c2  =  a2  +  b2. 

Hence,  c  can  not  be  less  than  &,  though  it  may  approach 
infinitely  near  to  the  value  of  a,  as  b  diminishes  toward 
zero.  Therefore, 

Theorem  II,  —  The  foci  of  any  hyperbola  fall  without 
the  curve. 

451.  Moreover,  c  —  a  measures  the  distance  of  either 
focus  from  the  adjacent  vertex;  while  the  distance  of 
either  from  the  remote  vertex  =  c  -f-  a.  Hence, 

Theorem  III.  —  The  vertices  of  the  curve  are  equally 
distant  from  the  foci. 


From  Art.  448,  the  length  of  the  transverse 
axis  =  2a.  But  (Art.  167)  2a  =  the  constant  difference 
of  the  focal  radii  of  any  point  on  the  curve.  That  is, 

Theorem  IV,  —  The  difference  of  the  focal  radii  of  any 
point  on  an  hyperbola  is  equal  to  the  length  of  its  trans- 
verse axis. 

Corollary.  —  We  may  therefore  construct  the  curve  by 
points  as  follows:  —  From  either  focus,  as  F',  lay  off 


366 


ANALYTIC  GEOMETRY. 


F'M  equal  to  the  transverse  axis.     Then  from  F'  as  a 

center,  with  any  radius  F1  R  greater  than  F'A,  describe 

two  small  arcs,  one  above  the 

axis,  and  the  other  below  it. 

From  the  remaining  focus  F 

as  a  center,  with  a  radius  MR, 

describe  two  other  arcs,  inter- 

secting the  former  in  P  and 

Pr  :  these  points  will  be  upon 


the  required  hyperbola;  for  F'P—FP=F'R  —  MR  = 
A'  A  =  F'P'  —  FP1.  By  using  the  radius  F'R  from  F, 
and  MR  from  Ff,  two  points,  P"  and  P'",  may  be  found 
upon  the  second  branch  of  the  curve.  The  operation  must 
be  repeated  until  the  outline  of  the  two  branches  is  dis- 
tinctly marked,  when  the  curve  may  be  drawn  through 
the  points  determined.  The  conjugate  curve  may  be 
formed  in  the  same  way,  at  the  same  time,  if  desired. 

453.   The  abbreviation  b2=c2  —  a2  adopted  (Art.  167) 
for  the  Hyperbola,  gives  us 


Hence,  attributing  to  a,  b,  c  the  meanings  now  known  to 
belong  to  them,  we  have 

Theorem  V. —  The  conjugate  semi-axis  of  any  hyperbola 
is  a  geometric  mean  between  the  segments  formed  upon  the 
transverse  axis  by  either  focus. 

Corollary. — Transposing  in  the 
abbreviation  above,  we  get  c2  — 
a2  -f  b2.  But,  from  the  diagram, 
d>  +  b2  =  AB*.  Therefore,  The 
distance  from  the  center  to  either 

focus  of  an  hyperbola  is  equal  to  the  distance  between  the 
extremities  of  its  axes.     Hence,  when  the  axes  are  given, 


LATUS  RECTUM  OF  HYPERBOLA.  367 

we  may  construct  the  foci  as  follows  :  —  From  the  center  (7, 
with  a  radius  equal  to  the  diagonal  of  the  rectangle  under 
the  semi-axes,  describe  an  arc  cutting  the  transverse  axis 
produced  in  F  and  Ff  :  the  two  points  of  intersection  will 
be  the  foci  sought. 

454.  By  an  analysis  similar  to  that  of  Art.  357,  the 
details  of  which  the  student  must  supply  ,*  we  obtain 

Theorem  VI.  —  The  squares  on  the  ordinates  drawn  to 
either  axis  of  an  hyperbola  are  proportional  to  the  rect- 
angles under  the  corresponding  segments  of  that  axis. 

Corollary.  —  For  the  ordinate  passing  through  either 
focus,  we  shall  therefore  have 


But  cz  —  a2=b2.    Hence,  doubling 

FP  or  F'P', 

2b2 

latus  rectum  =  —  =  . 

a  2a 

That  is,  The  latus  rectum  of  any   hyperbola  is  a   third 
proportional  to  the  transverse  axis  and  the  conjugate. 

455.  Throwing  the  equations  to  the  Hyperbola  and 
its  conjugate  into  the  forms 

y2  _  52  x2  _  a2 

(x+a)(x-a)  =  If  '       (y  +  b)(y-b)  =  ?  ' 

we  at  once  obtain 

Theorem  VII.  —  The  squares  on  the  axes  of  any  hyperbola 
are  to  each  other  as  the  rectangle  under  any  two  segments 
of  either  is  to  the  square  on  the  ordinate  which  forms  the 
segments. 


*  When  seeking  properties  of  the  conjugate  axis,  we  must  of  course  use 
the  equation  to  the  conjugate  hyperbola. 

An.  Ge.  34. 


368 


ANALYTIC  GEOMETRY. 


45f5o   If  we  put  the  equation  to  the  Hyperbola  into 
the  form 


and  compare  it  with  that  of  the  circle  described  upon 
the  transverse  axis,  namely,  with 


we  see  that  the  ordinates  of  the  two  curves,  correspond- 
ing to  a  common  abscissa,  have  only  an  imaginary  ratio. 
The  analogy  between  the  Hyperbola  and  the  Ellipse,  so 
far  as  concerns  the  circle  mentioned,  is  therefore  defective. 
If,  however,  we  suppose  b  =  a  in  (1),  we  get 

f  =  x--<f-  (2), 

and,  if  we  now  divide  (1)  by  (2),  we  obtain 
yh  :yr::b  :  a. 

Now  equation  (2)  evidently  represents  the  curve  which 
in  Art.  177  we  named  a  rectangular  hyperbola,  but 
which  we  may  henceforth  call  an  equilateral  hyperbola, 
since  its  equation  is  obtained  from  that  of  the  ordinary 
curve  by  supposing  the  axes  equal.  We  have  therefore 
proved 

Theorem  VIII.  —  The  ordinate  of  any  hyperbola  is  to  the 
corresponding  ordinate  of  its  equilateral,  as  the  conjugate 
semi-axis  is  to  the  semi-transverse. 

Remark  —  The  peculiarity  in  the  figure 
of  the  Equilateral  Hyperbola  is,  that  the 
curve  is  identical  inform  with  its  conjugate. 
For  the  equation  to  its  conjugate  (Art.  449) 
is 


and  if  we  transform  this  to  the  conjugate 
axis  as  the  axis  of  x,  by  revolving  the 


EQUILATERAL  HYPERBOLA. 


369 


reference-axes  through  90°,  and  therefore  (Art.  56,  Cor.  3)  replacing 
x  b  j  —  y,  and  y  by  x,  we  obtain 

a;2  —  y2=a2; 

so  that  the  conjugate  curve,  when  referred  to  its  own  transverse 
axis,  is  represented  by  the  same  equation  as  its  primary,  and  is 
therefore  the  same  curve.  The  diagram  presents  a  pair  of  conjugate 
equilaterals. 

Corollary.  —  Notwithstanding  the  defective  analogy  be- 
tween the  Ellipse  and  the  Hyperbola  with  respect  to  the 
circle  formed  upon  the  transverse  axis,  this  curve  still 
aids  us  in  fixing  the  meaning  of  the  abbreviation 

a*+V_ 
~~ 


adopted  in  Art.  170,  and  warrants  us  in  calling  e  the 
eccentricity  of  the  Hyperbola.  For,  as  in  the  case  of  the 
Ellipse,  since  a2  -j-  b2  =  c2,  we  learn  that  e  is  ike  ratio 
which  the  distance  from  the  center  to  either  focus  of  an 
hyperbola  bears  to  its  transverse  semi-axis.  Let  us,  then, 
suppose  a  series  of  ellipses  a'hd  hyperbolas  to  be  described 
upon  a  common  transverse 
axis  :  we  saw  (Art.  359, 
Cor.  2)  that,  as  the  vary- 
ing ellipse  of  such  a  series 
deviates  more  and  more 
from  the  circle  formed 
upon  the  same  axis,  and 

approaches  nearer  and  nearer  to  coincidence  with  the 
axis  A'  A,  the  eccentricity  e  advances  nearer  and  nearer 
to  the  limit  1.  Assuming,  then,  that  e  actually  reaches 
this  limit,  the  corresponding  ellipse  must  vanish  into  the 
line  A'  A,  which  forms  the  common  axis.  Now,  from  the 
abbreviation  above,  the  e  of  the  Hyperbola  lies  between 
the  limits  1  and  oo  :  hence  the  series  of  hyperbolas  may 
be  said  to  arise  out  of  the  common  axis  A1  A  at  the 


370  ANALYTIC   GEOMETRY. 

instant  when  the  series  of  ellipses  vanishes  into  it,  and 
to  recede  farther  and  farther  from  the  axis  as  e  advances 
from  1  toward  oo.  But  since  e,  with  respect  to  the  ellipses 
and  the  hyperbolas  both,  is  at  all  times  the  ratio  between 
the  same  elements  of  the  curves  ;  and  since,  taking  the  circle 
described  upon  the  common  axis  as  the  starting-point,  this 
ratio  steadily  advances  from  0  through  1  toward  oo;  we 
may  regard  the  recession  of  the  hyperbolas  from  the  axis 
A'  A  as  a  farther  deviation  from  the  curvature  of  the  circle 
mentioned,  and  consequently  call  <?,  which  measures  this 
deviation,  the  eccentricity. 

We  may  therefore  interpret  the  name  hyperbola  (derived 
from  the  Greek  bxepfldtt&v,  to  exceed)  as  signifying,  that, 
in  this  curve,  the  eccentricity  is  greater  than  unity. 

Since  e  increases  as  b  increases,  it  follows  that  the 
greater  the  eccentricity,  the  more  obtuse  will  be  the 
branches  of  the  corresponding  hyperbola.  In  case  the 
curve  is  equilateral,  or  b  -•=  a,  we  shall  have 


457.  The  distance  of  any  point  on  an  hyperbola  from 
either  focus,  may  be  expressed  in  terms  of  the  abscissa 
of  the  point.  For,  putting  p  to  denote  any  such  focal 
distance,  we  have  (Art.  167) 


Substituting  for  y"2  from  the  equation  to  the  curve,  and 
reducing  by  means  of  the  relations  in  Art.  171,  we  get 

p  =  ex  ±  a, 

where  the  upper  sign  corresponds  to  the  left-hand  focus, 
and  the  lower  to  the  right-hand  one.     Hence, 

Theorem  IX.  —  The  focal  radius  of  any  point  on  an 
hyperbola  is  a  linear  function  of  the  corresponding  abscissa. 


DIAMETERS.  371 

Remark. — The  expression  obtained  in  this  article,  like 
that  found  for  the  Ellipse  in  Art.  360,  is  accordingly 
known  as  the  Linear  Equation  to  the  Hyperbola. 


By  reasoning  similar  to  that  employed  in  Art. 
361,  we  may  verify  the  figure  of  the  Hyperbola,  as  drawn 
in  Art.  165.  We  leave  the  student  to  show,  by  interpret- 
ing the  equation 

2  V         2  2, 

that  the  curve  consists  of  two  infinite  branches,  separated 
by  the  transverse  axis  =  2a,  facing  in  opposite  directions, 
and  symmetric  to  both  axes. 

DIAMETERS. 

459.  Equation  to  any  Diameter. — To  obtain  an 
expression  for  the  locus  of  the  middle  points  of  chords 
in  an  hyperbola  which  have  a  common  inclination  6f  to 
the  transverse  axis,  we  write  (Art.  167,  Cor.)  —  b2  for  62 
in  the  final  equation  of  Art.  362.     Hence,  the  required 
equation  is 

y  =  -^.x  cot  6'. 

a 

Corollary. — Putting  6  —  the  inclination  of  the  diameter 
itself,  we  obtain  (Art.  78,  Cor.  1),  as  the  condition  con- 
necting the  inclination  of  any  diameter  with  that  of  the 
chords  which  it  bisects, 

tan  0  tan  0'=- -  . 
a2 

460.  Since  the  equation  to  a  diameter  conforms  to 
the  type  y  =  mx,  we  at  once  infer 

Theorem  X. — Every  diameter  of  an  hyperbola  is  a  right 
line  passing  through  the  center. 


372  ANALYTIC   GEOMETRY. 

Corollary. — The  angle  6'  being  arbitrary,  it  follows 
from  the  above  condition,  that  6  is  also  arbitrary. 
Hence  the  converse  theorem :  Every  right  line  that 
passes  through  the  center  of  an  hyperbola  is  a  diameter. 

4GI»   Eliminating,  then,  between  the  equation 
y  —  x  tan  0 

and   the   equation   to   the   Hyperbola,   we   get,    for  the 
abscissas   of   intersection   between  the    curve    and   any 

diameter, 

ab 


~  y(b2  —  a2tan2#) 

Now  these  abscissas  evidently  become  imaginary  when 
a2  tan2  6  >  b2.  Hence, 

Theorem  XI, —  The  proposition  that  every  diameter  cuts 
the  curve  in  two  real  points,  is  not  true  of  the  Hyperbola. 

Corollary  1. — It  is  obvious,  however,  that  the  intersec- 
tions will  be  real,  and  at  a  finite  distance  from  the  center, 
so  long  as  a2  tan2  0  <  b'2.  Hence,  the  diameters  corre- 
sponding to  a2  tan2  6  =  b2,  that  is,  the  two  diameters 
whose  tangents  of  inclination  are  respectively 

b  b 

tan  o  r=  —  ,      tan  u  = , 

a  a 

form  the  limits  between  those  diameters  which  have  real 
intersections  with  the  curve  and  those  which  have  not. 
But,  from  the  values  of  their  tangents  of  inclination, 
these  two  diameters  are  the  diagonals  of  the  rectangle 
contained  by  the  axes.  We  learn,  then,  that  diameters 
which  cut  the  Hyperbola  in  real  points  must  either 
make  with  the  transverse  axis  an  angle  less  than  is  made 
by  the  first  of  these  diagonals,  or  greater  than  is  made  by 
the  second. 


LENGTH  OF  DIAMETER.  373 

It  deserves  notice,  that  the  condition  a2  tan2  0  =  b2 
renders  the  abscissas  of  intersection,  as  expressed  above, 
infinite.  The  two  limiting  diameters  therefore  meet  the 
curve  at  infinity:  and  we  have  come  upon  the  analogue 
of  the  equi-conjugates  in  the  Ellipse.  We  shall  soon 
find  that  these  lines  are  the  most  remarkable  elements 
of  the  Hyperbola,  giving  it  a  series  of  properties  in 
which  the  other  Conies  do  not  share. 

Corollary  2. — Eliminating  between  y  =  x  tan  6  and  the 
equation  to  the  conjugate  hyperbola,  we  get 

L  ab 

~  y  (a2  tan2  #  —  62)  ' 

Here,  then,  the  condition  of  real  intersection  is  #2tan2#>62. 
Hence,  Every  diameter  that  cuts  an  hyperbola  in  two  imag- 
inary points,  cuts  its  conjugate  in  two  real  ones. 

462.  Length  of  any  Diameter. — This  being  double 
the  central  radius  vector  of  the  curve,  may  be  determined 
(Art.  170)  by  J2 

"    ~  e2  cos2  6  —  1 ' 

or,  if  the  diameter  meets  the  conjugate  curve  instead  of 
the  primary,  by  72 

P     =  1  —  e2'cos2  6  ' 

an  expression  readily  obtained  by  transforming  to  polar 
co-ordinates  (Art.  57,  Cor.)  the  equation  to  the  conjugate 
hyperbola,  found  in  Art.  449. 

463.  The  first  of  the  above  expressions  is  least  when 
0  =  0 ;  and  the  second,  when  0  =  90°.     Hence, 

Theorem  XII. — Each  axis  is  the  minimum  diameter  of 
its  own  curve. 

Remark. — We  see,  then,  that  the  terras  major  and  minor 
are  not  applicable  to  the  axes  of  an  hyperbola. 


374  ANALYTIC  GEOMETRY. 

464.  By  the  same  argument  as  in  Art.  367,  we  obtain 

Theorem  XIII, — Diameters  which  make  supplemental 
angles  with  the  transverse  axis  of  an  hyperbola  are  equal. 

Corollary. — It  also  follows,  as  in  the  corollary  to  Art. 
367,  that  we  can  construct 
the  axes  when  the  curve  is 
given.  The  diagram  illustrates 
the  process  in  the  case  of  the 
Hyperbola,  and  the  student 
may  transfer  the  statements 
of  Art.  367,  Cor.,  to  this  figure, 
letter  by  letter.  We  deem  it  unnecessary  to  repeat  them. 

465.  The  inclinations  of  two  diameters  being  repre- 
sented by  6  and  6',  the  argument  of  Art.  368  obviously 
applies  to  the  Hyperbola,  with  respect  to  the  condition 
(Art,  459,  Cor.) 

tan/?  tan  0'=—  - 
Hence, 

Theorem  XIV. — If  one  diameter  of  an  hyperbola  bisects 
chords  parallel  to  a  second,  the  second  bisects  chords  par- 
allel to  the  first. 

466.  Two  diameters  of  an  hyperbola  which  are  thus 
related,  are  called  conjugate  diameters,  as  in  the  case  of 
the  Ellipse.     The  phrase  ordinates  to  any  diameter  is 
also  used  in  connection  with  the  Hyperbola,  to  signify 
the  halves  of  the  chords  which  the  diameter  bisects ;  or, 
the  right  lines  drawn  from  the  diameter,  parallel  to  its 
conjugate,  to  meet  the  curve. 

Corollary. — The  construction 
of  a  pair  of  conjugates  in  an 
hyperbola,  may  therefore  be 
effected  exactly  in  the  manner 


CONJUGATE  DIAMETERS.  375 

described  in  the  corollary  to  Art.  369.  The  details  may 
be  gathered  by  applying  to  the  parts  of  the  annexed 
diagram,  the  statements  of  that  corollary. 

467.  Equation  of  Condition  for  Conjugates  in 
the  Hyperbola. — The  conjugate  of  any  diameter  being 
parallel  to  the  chords  which  the  diameter  bisects,  the  in- 
clinations of  two  conjugates  must  be  connected  in  the 
same   way   as   those   of  a   diameter   and   its   ordinates. 
Hence,  if  6  and   0'  represent  the  inclinations,  the  re- 
quired condition  (Art.  459,  Cor.)  is 

52 

tan  0  tan  0'=-,. 

a2 

Corollary, — Hence,  if  tan  0  <  b  :  a,  tan  61  >  b  :  a  ;  and 
if  tan  6  >  —  I  :  a,  tan  0'  <  —  I :  a.  Therefore  (Art.  461, 
Cors.  1,  2),  If  one  of  two  conjugates  meets  an  hyperbola, 
the  other  meets  the  conjugate  curve. 

Remark. — The  condition  of  this  article  might  have  been  obtained 
from  that  of  Art.  370,  by  merely  changing  b2  into  —  62. 

468.  The  preceding  condition  shows  that  the  tangents 
of  inclination  have  like  signs.     Hence,  the  angles  made 
with  the  transverse   axis  by  two  conjugates  are  either 
both  acute,  or  else  both  obtuse.     That  is, 

Theorem  XV. —  Conjugate  diameters  of  an  hyperbola  lie 
on  the  same  side  of  the  conjugate  axis. 

469.  Equation  to  a  Diameter  conjugate  to  a 
Fixed  Point. — Making  the  requisite   change   of   sign 
(Art.  167,  Cor.)  in  the  equation  of  Art.  372,  we  get  the 
one  now  sought,  namely, 


An.  Ge.  35 


376  ANALYTIC   GEOMETRY. 

Corollary.  —  The  diameter  conjugate  to  that  which 
passes  through  (a,  0)  is  therefore  x  =  0,  that  is,  the 
conjugate  axis.  Hence,  The  axes  of  an  hyperbola  con- 
stitute a  case  of  conjugate  diameters. 

47O.  Problem,  —  Given  the  co-ordinates  of  the  extrem- 
ity of  a  diameter,  to  find  those  of  the  extremity  of  its 
conjugate. 

By  the  extremities  of  the  conjugate  diameter,  are 
meant  the  points  in  which  the  conjugate  cuts  the  conju- 
gate hyperbola.  The  required  co-ordinates  are  therefore 
found  by  eliminating  between  the  equation  of  Art.  469 
and 


They  are 


Remark,  —  By  comparing  these  expressions  with  those 
of  Art.  373,  we  notice  that  the  abscissa  and  ordinate  of 
the  conjugate  diameter  in  the  Ellipse  have  opposite  signs, 
but  in  the  Hyperbola  like  signs.  This  agrees  with  the 
properties  developed  in  Arts.  371,  468. 

471.  The  equations  of  Art.  470,  like  those  of  Art.  373,  give 
rise  to 

Theorem  XVI.  —  The  abscissa  of  the  extremity  of  any  diameter  is 
to  the  ordinate  of  the  extremity  of  its  conjugate,  as  the  transverse  axis 
is  to  the  conjugate  axis. 

472.  By  following,  with  respect  to  the  second  ex- 
pression of  Art.  470,  the   steps  indicated  in  Art.  375, 
excepting  that  we  subtract  the  ?//2,  we  arrive  at 

Theorem  XVII,  —  The  difference  of  the  squares  on  the 
ordinates  of  the  extremities  of  conjugate  diameters  is  con- 
stant,  and  equal  to  the  square  on  the  conjugate  semi-axis. 


LENGTH  OF  CONJUGATES.  377 

Remark.  —  We  leave  the  student  to  prove  the  analogous 
property  :  The  difference  of  the  squares  on  the  abscissas 
of  the  extremities  of  conjugate  diameters  is  constant,  and 
equal  to  the  square  on  the  transverse  semi-axis. 

473.  Problem,  —  To  find  the  length  of  a  diameter  in 
terms  of  the  abscissa  of  the  extremity  of  its  conjugate. 

Let  x'yr  be  the  extremity  of  any  diameter,  a'  half  its 
length,  and  b'  half  the  length  of  its  conjugate.  Then 
a'2  =  x'2+y'\  and  we  get  (Art.  470) 


since  xc  and  yc  must  satisfy  the  equation  to  the  conjugate 
hyperbola.     Hence,  (Art.  171,) 

a''2  =  e2x2  -f-  a2. 

By  performing  similar  operations  with  respect  to  bf,  we 
should  get 

b'2  =  e2x.'2—a2. 

474.  Between  these  results  and  those  of  Art.  376, 
there  is  a  striking  difference  ;  and,  as  only  the  value  of 
I'2  equals  (Art.  457)  the  rectangle  of  the  focal  radii 
drawn  to  x'y',  it  appears  as  if  the  property  proved  of 
the  Ellipse  in  Art.  377  were  only  true  of  the  Hyperbola 
with  respect  to  those  diameters  which  meet  the  conju- 
gate curve  instead  of  the  primary.  But  when  we  reflect 
that  it  is  entirely  arbitrary  which  of  two  conjugate  hyper- 
bolas we  consider  the  primary,  it  becomes  evident  that 
the  property  of  Art.  377  is  also  true  of  the  diameters 
which  meet  the  curve  hitherto  called  the  primary,  pro- 
vided we  suppose  the  focal  radii  in  question  to  be  drawn 
from  the  foci  of  the  conjugate  curve.  With  this  under- 
standing, then,  we  may  state 


378  ANALYTIC  GEOMETRY. 

Theorem  XVIII.  —  The  square  on  any  semi-diameter  of 
an  hyperbola  is  equal  to  the  rectangle  under  the  focal  radii 
drawn  to  the  extremity  of  its  conjugate. 

47*5.  It  is  evident  on  inspection,  that  the  fourth  formula  in 
Art.  378  will  not  be  altered  by  changing  62  into  —  b2.  Hence,  in 
the  Hyperbola  as  well  as  in  the  Ellipse,  we  have 


(a  —  ex' 


in  which  x'y*  is  the  extremity  D  of 
any  diameter  IYD.  But  xn  +  y/2  = 
a'2=  (Art.  473)  eV  +  a2=  (Art. 
472,  Rem.  )  &\x/'1—  a2)  +  a2.  Hence, 
after  substituting  and  reducing, 


or,  in  the  Hyperbola  as  well  as  in  the  Ellipse,  we  have 

Theorem  XIX.  —  The  distance  from  the  extremity  of  any  diameter 
to  its  conjugate,  measured  upon  the  corresponding  focal  radius,  is 
constant,  and  equal  to  the  transverse  semi-axis. 

476.  Let  a',  V  denote  the  lengths  of  any  two  conju- 
gate semi-diameters  in  an  hyperbola.  Then  (Art.  473) 


'2 


(1). 


Also,  b'2  =  x*  +  y2  =  xc2  +  {  b2  (x2  -f  a2)  :  a2  }  ,  since  z,  and 
yc  satisfy  the  equation  to  the  conjugate  hyperbola. 
Hence,  (Art.  171,) 

6/2  =  62X2  +  £2  (2). 

Subtracting  (2)  from  (1),  member  by  member, 
a12  —  bf2  =  a2  —  b2. 

Hence,  as  the  antithesis  of  Art.  379, 

Theorem  XX.  —  The  difference  of  the  squares  on  any 
two  conjugate  diameters  of  an  hyperbola  is  constant  ,  and 
equal  to  the  difference  of  the  squares  on  the  axes. 


PARALLELOGRAM  OF  CONJUGATES. 


379 


477.  Angle  between  two  Conjugates.  —  Using  the 
same  symbols  as  in  Art.  380,  it  is  plain  that,  in  the 
Hyperbola  also,  we  shall  have 


Substituting  for  xc  and  yc  from  Art.  470,  reducing,  and 
remembering  that  b2xf2  —  a?y'2  =  a2b2,  we  get 

ab 


478.  Clearing  this  expression  of  fractions,  we  have 
a'b'  sin  <p  —  ab. 

The  first  member  of  this  equa- 
tion  obviously  expresses  the  area 
of  the  parallelogram  CDRS;  and 
the  second,  that  of  the  rectangle 
CAQB.  Therefore, 

Theorem  XXL  —  The  parallelogram  under  any  two  con- 
jugate diameters  is  constant,  and  equal  to  the  rectangle 
under  the  axes. 

Remark.  —  The  diagram  represents  the  parallelogram  and  rectangle 
as  inscribed  in  the  pair  of  conjugate  hyperbolas.  The  figure  will  in 
due  time  be  justified.  Also,  as  in  the  case  of  the  Ellipse,  the  the- 
orem might  have  been  stated  thus  :  The  triangle  formed  by  joining 
the  extremities  of  any  two  conjugate  diameters  is  constant,  and  equal  to 
that  included  between  the  semi-axes. 

Corollary  1.  —  If  we  suppose  <p  =  90°,  then  sin  <p  =  1  ; 
and  we  get 

a'b'  =  ab. 
Now  (Art.  476), 


380  ANALYTIC  GEOMETRY. 

Solving  these  equations  for  af  and  b',  we  find,  as  the  only 
real  values, 

a'  =  a,     bf=b. 

Therefore,  In  any  hyperbola  there  is  but  one  pair  of  con- 
jugate diameters  at  right  angles  to  each  other,  namely,  the 
axes. 

Corollary  2.— We  saw  (Arts.  381,  Cor.  1;  382,  Cor.) 
that,  in  the  Ellipse,  sin  </?  lies  between  the  limits  1  and 
2ab  :  (a2-h&2).  But  (Art.  476),  a'2  =  b'2  +  constant,  in  the 
Hyperbola:  whence  a'  and  b'  must  increase  or  diminish 
together.  Therefore,  as  any  diameter  (Art.  461,  Cor.  1) 
tends  toward  an  infinite  length  the  nearer  its  inclination 
approaches  the  limit  6  =  tan"1  b  :  a,  the  semi-conjugates 
a'  and  b'  must  advance  together  toward  the  value  cc^  and 
the  product  a'br  tends  toward  GO  for  its  maximum.  That 
is,  sin  <p  tends  toward  the  limit  0 ;  or,  The  angle  between 
two  conjugates  in  an  hyperbola  diminishes  without  limit. 

But  though  the  conjugates  thus  tend  to  final  coinci- 
dence as  each  tends  to  an  infinite  length,  the  relation 
a'2  —  b12  =  constant  renders  it  impossible  that  the  condi- 
tion a'  =  b'  shall  ever  arise  in  the  Hyperbola,  unless 
the  curve  is  equilateral.  The  infinite  diameters  that 
form  the  limit  of  the  ever-approaching  conjugates  are 
therefore  not  equal  infinites,  and  the  conception  of 
equi-conjugates  is  not  in  general  present  in  the  curve. 
However,  from  the  equation  of  condition  for  conjugate 
diameters,  namely, 

b2 

tan#  tan  6'  =  - •  , 
a2 

it  is  plain  that  when  the  conjugates  finally  coincide, 
each  makes  with  the  transverse  axis  an  angle  whose 
tangent  is  either  b  :  a  or  else  — b  :  a.  Hence,  the  two 


THE  SELF-CONJUGATE  DIAMETERS.  381 

right    lines    which    pass    through   the    center   with    the 
respective  inclinations 

-1  -1 


, 
a  a 


may  each  be  regarded  as  the  limiting  case  of  a  pair  of 
conjugate  diameters  ;  or,  each  may  be  called  a  diameter 
conjugate  to  itself.  The  curve,  then,  replaces  the  con- 
ception of  equi-conjugates  by  that  of  self-conjugates. 

479.  From  what  has  just  been  shown,  it  follows  that 
the  inclinations  of  the  self-conjugate  diameters  to  the 
transverse  axis  are  determined  by  the  formula 

tan  6  =  ±  -  • 
a 

By  drawing  the  rectangle  of  the 

axes,  LMNR,  it  becomes  evi- 

dent that  the  first  of  the  values 

here  expressed  corresponds  to  the  angle  ACL;  and  the 

second,  to  the  angle  ACM.     Hence, 

Theorem  XXII.  —  The  self  -conjugates  of  an  hyperbola 
are  the  diagonals  of  the  rectangle  contained  under  its  axes. 

Corollary.  —  Hence,  further,  An  hyperbola  has  two,  and 
only  two,  self  -conjugates.  Their  mutual  inclination  LCM, 
or  LCR,  as  we  readily  find,  is  determined  by 


4SO.  We  have  thus  found  the  two  lines  of  the  Hyper- 

bola which,  in  Art.  383,  we  said  were  foreshadowed  by 
the  equi-conjugates  of  the  Ellipse.  That  the  two  self- 
conjugates  are  in  reality  the  analogue  of  the  equi- 


382  ANALYTIC   GEOMETRY. 

conjugates,  we  can  easily  show :  for  though  it  is  true, 
as  we  saw  in  the  second  corollary  to  Art.  478,  that  the 
two  infinitely  long  conjugates  which  unite  in  either  of 
the  self-conjugates  are  not  equal  infinites,  still  the  two 
self-conjugates,  when  compared  with  each  other,  are 
equal  infinites.  For,  since  they  make  equal  angles 
ACL,  ACR  with  the  transverse  axis,  they  are  the 
limiting  case  to  which  two  equal  diameters  DP,  D'Q 
necessarily  tend  as  their  extremities  D  and  D1  move 
along  the  curve  in  opposite  directions  from  the  vertex  A. 
But  the  chief  interest  of  the  self-conjugates  is  due  to 
a  property  in  which  the  equi-conjugates  of  the  Ellipse 
have  no  share,  and  in  virtue  of  which  they  are  called 
the  asymptotes  of  the  Hyperbola.  From  this  property 
are  derived  several  others,  peculiar  to  the  latter  curve, 
which  will  receive  a  separate  consideration  in  the  proper 
place. 

THE    TANGENT. 

481.  Equation  to  the  Tangent. — To  obtain  this 
for  the  Hyperbola,  we  simply  change  b2  into  —  b2  in  the 
equation  of  Art.  385.  We  thus  get 

x'x       y'y  _ 
~d2          h*    ' 

4S2.  Condition  that  a  Sight  Line  shall  touch 
an  Hyperbola. — Making  the  characteristic  change  of 
sign  in  the  condition  of  Art.  386,  we  have 


as  the  condition  that  the   line  y  —  mx  -}-  n  may  touch' 
the  curve 


ECCENTRIC  ANGLE.  383 

Corollary.  —  Hence,  every  line  whose  equation  is  of  the 
form 


y  =  mx  -\-  ymPa*  —  2 

is  a  tangent  to  the  hyperbola  whose  semi-axes  are  a  and 
b.  Like  the  similar  expressions  found  in  treating  the 
Circle  and  the  Ellipse,  an  equation  of  this  form  is  called 
the  Magical  Equation  to  the  Tangent. 


The  Eccentric  Angle. — The  expression  of  any  point  on 
an  hyperbola  in  terms  of  a  single  variable,  is  effected  by  employing 
an  angle  analogous  to  that  whose 
use  in  connection  with  the  Ellipse 
was  explained  in  Art.  387.  If  from 
the  foot  of  the  ordinate  correspond- 
ing to  any  point  P  of  an  hyperbola,  >/  \^ 
we  draw  M Q  tangent  to  the  inscribed 
circle  at  Q,  and  join  Q  to  the  center  C,  QCM  is  called  the  eccentric 
angle  of  P. 

Now  (Trig.,  860)  CM=CQ  sec  QCM.  Also,  from  the  equation 
to  the  Hyperbola,  combined  with  this  value  of  CM, 

MP2  =  ~(  CM2  -  a2)  =  &»  tan2  Q  CM. 

Hence,  if  we  represent  the  arbitrary  point  P  by  vftf, 

y/  =  a  sec  0,      y'  =  I  tan  9. 

Substituting  for  x/  and  y'  in  Art.  481,  we  may  write  the  equation 
to  the  tangent,  in  this  notation, 

x  y   . 

—  sec  0  —  V  tan  0  —  1. 

a  b 

The  analogy  of  the  angle  QCJJ/,  as  formed  in  the  case  of  the 
Hyperbola,  to  the  similarly  named  angle  in  the  Ellipse,  may  perhaps 
be  obscure  to  the  beginner;  but  it  will  become  apparent  when  we 
reach  the  conception  of  a  hyperbolic  subtangent. 


•384  ANALYTIC  GEOMETRY. 

484.  Problem. — If  a  tangent  to  an  hyperbola  passes 
through  a  fixed  point,  to  find  the  co-ordinates  of  contact. 

Let  x"y"  be  the  fixed  point,  and  x'y'  the  required 
point  of  contact.  Then,  changing  the  sign  of  b2  in  the 
results  of  Art.  388,  we  get 


,  _  a2b2x"  g=  aY  Va2y"2  -  b2x"2  -f-  a2b2 
tfxn2  —  a2ym  ~  ' 


f_a2b2if±b2xnVa2y"2  -b2x"2-}-  a2b2 

y    - 


Corollary  1.  —  The  form  of  these  values  indicates  that 
from  any  given  point  two  tangents  can  be  drawn  to  an 
hyperbola  :  real  when  a?y"2  —  b2x"2  -f-  a2b2  >  0,  that  is, 
when  the  point  is  inside  of  the  curve  ;  coincident  when 
a2y"2  —  b2x"2  -f-  a2b2  =  0,  that  is,  when  the  point  is  on  the 
curve  ;  imaginary  when  a2y"  —  b2x"2  -\-  a2b2  <  0,  that  is, 
vdien  the  point  is  outside  of  the  curve. 

Corollary  2.  —  With  regard  to  any  two  real  tangents 
drawn  from  a  given  point,  it  is  evident  that  their  ab- 
scissas of  contact  will  have  like  signs,  if  they  both  touch 
the  same  branch  of  the  curve,  and  unlike  signs,  if  the 
two  touch  different  branches.  But,  if  the  two  values  of 
x'  above  have  like  signs,  then,  merely  numerical  relations 
being  considered, 

a2b2xn  >  a2y"  I/a2/  f2  —  b  2x"2  +  a2b2  ; 
that  is,  after  squaring,  transposing,  and  reducing, 


Hence,  as  y  =  (b  :  a)  x  is  the  equation  to  the  diagonal 
of  the  rectangle  formed  upon  the  axes  (Art.  461,  Cor.  1), 


POSITION  OF  POINT  OF  CONTACT.  385 

the  ordinate  of  the  point  from  which  two  tangents  can 
be  drawn  to  the  same  branch  of  an  hyperbola  must  be 
less  than  the  corresponding  ordinate  of  the  diagonal; 
that  is,  the  point  itself  must  lie 
somewhere  within  the  space  in- 
cluded between  the  self-conjugates 
CL,  OR  (or  CM,  ON)  and  the 
adjacent  branch  of  the  curve. 
Hence,  generally,  The  two  tan- 
gents which  can  be  drawn  to  an  hyperbola  from  any 
point  inside  of  the  curve,  will  touch  the  same  branch  or 
different  branches,  according  as  the  point  is  taken  within 
or  without  the  angle  of  the  self -conjugates  which  incloses 
the  two  branches. 

485.  The  argument  of  Art.  889  will  be  seen,  on  a 
moment's  inspection,  to  hold  good  when  hyperbolic 
equations  are  substituted  for  the  elliptic.  Therefore, 

Theorem  XXIII. —  The  tangent  at  the  extremity  of  any 
diameter  of  an  hyperbola  is  parallel  to  the  conjugate 
diameter. 

Corollary. — Tangents  at  the  extremities  of  a  diameter 
are  parallel  to  each  other. 

Remark. — By  drawing  any  diameter  and  its  conjugate, 
and  passing  a  parallel  to  the  latter  through  the  extremity 
of  the  former,  we  can  readily  form  a  tangent  to  a  given 
hyperbola.  If  we  construct  tangents  at  the  extremities 
of  both  diameters,  we  shall  have  an  inscribed  parallelo- 
gram. Thus  the  diagram  of  Art.  478  is  verified;  for, 
as  only  one  parallel  to  a  given  line  can  be  drawn 
through  a  given  point,  lines  drawn  through  the  extremi- 
ties of  conjugate  diameters  so  as  to  form  their  parallelo- 
gram must  be  tangents  to  the  curve. 


386  ANALYTIC  GEOMETRY. 

486.  Let  PT  be  a  tangent  to  an  hyperbola  at  any 
point  P,  and  FP,  F'P  its  focal  radii  of  contact.  From 
the  equations 

,8 

Ftfx  —  a2y'y  =  a2b2  ( PT), 
y'(x-c)-(x'-c)y=Q  (FP), 
tf(x+c)-W+c)y=Q  (F'P), 

we  readily  find,  by  the  same  steps  as  in  Art.  390, 

tan  FPT  =  —. ,     tan  F'PT=  —, . 

cy'  cy' 

Hence,  FPT  =  F'PT-,  or,  we  have 

Theorem  XXIV. —  The  tangent  of  an  hyperbola  bisects 
the  internal  angle  between  the  focal  radii  drawn  to  the 
point  of  contact. 

Corollary  1. — We  therefore  obtain  the  following  solu- 
tion of  the  problem  :  To  construct  a  tangent  to  an  hyper- 
bola at  a  given  point.  Draw  the  focal  radii  FP,  F'P  to 
the  given  point  P.  On  the  longer,  say  F'P,  lay  off 
PQ  =  FP,  and  join  QF.  Through  P  draw  SPT  at 
right  angles  to  QF:  then  will  SPT  be  the  tangent 
sought.  For  QPF  is  by  construction  an  isosceles  tri- 
angle; and  SPT,  the  perpendicular  from  its  vertex  to 
its  base,  must  therefore  bisect  the  angle  F'PF. 

Corollary  2. — Hence,  all  rays  emanating  from  F,  and 
striking  the  curve,  will  be  reflected  in  lines  which,  if 
traced  backward,  converge  in  F';  and  reciprocally. 
Accordingly,  to  suggest  the  resemblance  between  these 
points  and  the  corresponding  ones  of  the  Ellipse,  they 
are  called  the  foci,  or  burning  points,  of  the  Hyperbola. 


PRINCIPAL  SUB  TANGENT. 


387 


487.  Let  us  suppose  y  =  0  in  the  equation 
b2xfx  —  a2y'y  =  a2b2. 

We  shall  thus  find,  as  the  value  of  the  intercept  which 
the  tangent  makes  upon  the 
transverse  axis, 


x=CT= 


x' 


In   the   Hyperbola,  then,  as 

well   as   in   the   Ellipse,  this 

intercept    is    a    third    proportional   to   the    abscissa    of 

contact  and  the  transverse  semi-axis,  and  we  have  the 

same  constructions  for  the  tangent  at  any  point  P  of 

the  curve,  or  from  any  point  T  of  the  transverse  axis, 

as  are  described  in  Art.  391. 

488.  The  Subtangent.— For  the  length  of  the 
subtangent  of  the  curve  in  the  Hyperbola,  we  have 
MT=CM—CT-,  or,  by  the  preceding  article, 


sub  tan  = 


x'2  — 


a) 


x'  x1 

But  x'  +  a  =  AM,  and  xr  —  a  =  MA.     Hence, 

Theorem  XXV. —  The  subtangent  of  an  hyperbola  is  a 
fourth  proportional  to  the  abscissa  of  contact  and  the  tivo 
segments  formed  upon  the  transverse  axis  by  the  ordinate 
of  contact. 

Corollary  1. — Let  xcr  be  the  abscissa  of  contact  for  any 
tangent  to  the  circle  described  on  the  transverse  axis  of 
an  hyperbola,  and  xhf  that  of  anytangent  to  the  hyperbola 
itself.  Then  (Art.  311), 


subtan  circ.  = 


388  ANALYTIC  GEOMETRY. 

Suppose,  now,  that  xc'  =  a2  :  xh'  ;  that  is  (Art.  487),  that 
the  abscissa  of  contact  in  the  circle  is  the  intercept  of  a 
tangent  to  the  hyperbola.  We  at  once  get 


subtan  circ.  =  —  -  r—  =  subtan  hyp. 
a* 

We  see,  then,  that  if  from  the  foot  T  of  any  tangent  to 

an  hyperbola  an  ordinate 

TQ  be  drawn  to  the  in- 

scribed circle,  the  tangent 

to  this  circle  at  Q  will  pass 

through  My  the  foot  of  the 

ordinate  of  contact  in  the 

hyperbola  ;  or,  If  tangents 

be  drawn  to  an  hyperbola  and  its  inscribed  circle  from  the 

head  and  foot  of  any  ordinate  to  either,  the  resulting  sub- 

tangents  will  be  identical. 

We  thus  learn  that  the  corresponding  points  of  an 
hyperbola  and  its  inscribed  circle  are  those  which  have  a 
common  subtangent.  And,  in  fact,  by  turning  to  the  dia- 
gram of  Art.  392,  it  will  be  seen  that  the  corresponding 
points  of  an  ellipse  and  its  circumscribed  circle  may  be 
defined  in  the  same  way.  Hence,  the  defect  in  the  anal- 
ogy between  the  two  curves  with  respect  to  those  circles, 
which  came  to  light  in  Art.  456,  can  now  be  supplied. 

Corollary  2,  —  Accordingly,  we  can  construct  the  tan- 
gent by  means  of  the  inscribed  circle  as  follows  :  —  When 
the  point  of  contact  P  is  given,  draw  the  ordinate  PM9 
and  from  its  foot  M  make  MQ  tangent  to  the  inscribed 
circle  at  Q.  Let  fall  tire  circular  ordinate  QT,  and  join 
its  foot  T  with  the  given  point  P.  PT  will  be  the 
required  tangent,  by  the  property  established. 

When  T  the  foot  of  the  tangent  is  given,  erect  the 


CENTRAL  PERPENDICULAR  ON  TANGENT.     389 

circular  ordinate  TQ,  and  draw  the  corresponding  tan- 
gent QM.  From  M9  the  foot  of  this,  erect  the  hyperbolic 
ordinate  MP,  and  join  its  extremity  P  with  the  given 
point  T. 

Remark.  —  By  comparing  the  diagrams  of  Arts.  387,  483  with 
those  of  Art.  392  and  the  present  article,  the  complete  analogy  of 
the  eccentric  angles  in  the  two  curves  will,  as  we  stated  in  Art.  483, 
become  apparent.  The  eccentric  angle  of  any  point  on  either  curve, 
may  be  defined  as  the  central  angle  determined  by  the  corresponding 
point  of  the  circle  described  upon  the  transverse  axis,  it  being  under- 
stood that  the  "corresponding"  points  are  those  which  have  a 
common  subtangent. 

489.  Perpendicular  from  the  Center  to  any 
Tangent.  —  The  length  of  the  perpendicular  from  the 
origin  upon  the  line 

b2x'x  —  a?y'y  =  a2b2, 
(Art.  92,  Cor.  2)  must  be 

a?b2  ab 


~ 


V  (6V2  -f  ay2)  ~    i/(e2x'2  —  a2)  ' 
Now  (Art.  473)  e2x'2—  a2=b'2.     Therefore, 

ab 
P  =  ¥> 

or,  as  in  the  Ellipse,  we  have 

Theorem  XXVI.  —  The  central  perpendicular  upon  any 
tangent  of  an  hyperbola  is  a  fourth  proportional  to  the 
parallel  semi-diameter  and  the  semi-axes. 

4OO.  Central  Perpendicular  in  terms  of  its  inclination 
to  tlie  Transverse  Axis.—  Changing  the  sign  of  &2  in  the  formula 
of  Art.  394,  we  get 


390  ANALYTIC  GEOMETRY. 

491o  Making  the  same  change  in  the  final  equation  of  Art.  395, 
we  obtain,  as  the  equation  to  the  locus  of  the  intersection  of  tangents 
to  an  hyperbola  which  cut  at  right  angles, 


From  this  (Art.  136)  we  at  once  get 

Theorem  XXVII, —  The  locus  of  the  intersection  of  tangents  to  an 
hyperbola  which  cut  each  other  at  right  angles,  is  the  circle  described 
from  the  center  of  the  hyperbola,  with  a  radius  =  Vv?  —  b\ 

492.  Perpendiculars  from  the  Foci  to  any 
Tangent. — For  the  length  of  the  perpendicular  from 
the  right-hand  focus  (ae,  0)  upon  b 2xfx  —  a2y'y  =  a2b2, 
we  have  (Art.  105,  Cor.  2) 

b2xfae  — •  a2b2  b(exf  —  a] 

~  y(b*x'L  +  ay2)  ~~  y  (e*x'2  —  a2)  ' 

or,  since  (Arts.  457,  473)  ex' — a  =  p,  and  e2x'2—a2=bf2, 

=  bp 
P       b'  ' 

And,  in  like   manner,   for   the  perpendicular  from  the 
left-hand  focus, 


Corollary.  —  Since  b'2  —  pp'  (Art.  474),  we  may  also 
write 

VP  ,2          bY 

~- 


493.  Upon  dividing  the  value  of  p  by  that  of  p',  we 
obtain 

Theorem  XXVIII.  —  The  focal  perpendiculars  upon  any 
tangent  of  an  hyperbola  are  proportional  to  the  adjacent 
focal  radii  of  contact. 


FOCAL  PERPENDICULARS  ON  TANGENT.       391 

And  if  we  multiply  these  values  together,  pp'  —  62; 
or,  \ve  have 

Theorem  XXIX. —  The  rectangle  under  the  focal  perpen- 
diculars upon  any  tangent  is  constant,  and  equal  to  the 
square  on  the  conjugate  semi-axis. 

494.  Changing  the  sign  of  b2  in  the  first  two  equa- 
tions of  Art.  399,  we  get 


y  —  mx  =  1/wiV  - 


my  -f-  x  =  Yd2  -f  b'\ 

as  the  equations  to  any  hyperbolic  tangent  and  its  focal 
perpendicular.  Adding  the  squares  of  these  together, 
we  eliminate  in,  and  obtain 

x2  +  y2  =  a2 

as  the  constant  relation  between  the  co-ordinates  of 
intersection  belonging  to  these  lines.  Hence,  (Art. 
136,) 

Theorem  XXX.— The  locus  of  the  foot  of  the  focal  per- 
pendicular upon  any  tangent  of  an  hyperbola,  is  the  circle 
inscribed  within  the  curve. 

Corollary. — We  may  therefore  apply  in  the  case  of 
the  Hyperbola,  the  construction  given  in  the  corollary 
to  Art.  399,  as  follows: 

To  draw  a  tangent  to  an 
hyperbola,  through  any  given 
point :  —  Join  the  given 
point  P  with  either  focus 
F,  and  upon  PF  describe  a 
circle  cutting  the  inscribed 
circle  in  Q  and  Q'.  The 
line  which  joins  P  to  either  of  these  points,  for  example 
An.  Ge.  36. 


392  ANALYTIC  GEOMETRY. 

the  line  PQ,  will  touch  the  hyperbola  at  some  point  T\ 
for  the  angles  PQF,  PQ'F  being  inscribed  in  a  semi- 
circle, Q  and  Q'  are  the  feet  of  focal  perpendiculars. 

When  P  is  on  the  curve,  and  PF  consequently  a 
focal  radius,  we  can  prove,  as  in  Ex.  8,  p.  359,  that 
the  circle  described  on  PF  will  touch  the  inscribed 
circle.  The  foot  of  the  focal  perpendicular  must  then 
be  found  by  joining  the  middle  point  of  PF  with  the 
center  (7,  and  noting  the  point  in  which  the  resulting 
line  cuts  the  inscribed  circle. 

493.  We  see,  then,  that  if  an  hyperbola  is  given,  every 
chord  drawn  from  the  focus  to  meet  the  inscribed  circle 
must  be  a  focal  perpendicular  to  some  tangent  of  the 
hyperbola.  On  the  other  hand,  it  is  obvious  that  any 
point  outside  of  a  given  circle,  may  be  considered  the 
focus  of  some  circumscribed  hyperbola.  Hence, 

Theorem  XXXI. — If  from  any  point  without  a  circle  a 
chord  be  drawn,  and  a  perpendicular  to  it  at  its  extremity, 
the  perpendicular  luill  be  tangent  to  the  circumscribed  hyper- 
bola of  which  the  point  is  a  focus. 

Corollary — Since  this  is  equivalent  to  saying  that  the  hyperbola 
is  the  envelope  of  the  perpendicular, 
we  may  approximate  the  outline  of 
an  hyperbola,  as  is  done  in  the  an- 
nexed figure,  by  drawing  chords  to 
a  circle  from  a  fixed  point  P  outside 
of  it,  and  forming  perpendiculars  at 
their  extremities.  It  should  be  no- 
ticed, that  only  the  parts  of  these 
perpendiculars  which  lie  on  opposite 
sides  of  the  chord  that  determines 
them,  enter  into  the  formation  of  the 
curve;  in  the  Ellipse,  on  the  contrary, 
the  perpendiculars  lie  on  the  same 
side  of  the  determining  chords.  When  the  chords  assume  the 


NORMAL  OF  THE  HYPERBOLA.  393 

limiting  positions  PL,  P72,  so  as  to  touch  the  circle  at  L  and  J?, 
the  corresponding  perpendiculars  LN,  MR  are  the  two  lines  which 
we  have  named  the  self-conjugates. 

4OG.  A  little  inspection  of  the  equations  in  Art.  401, 
after  the  sign  of  b2  has  been  changed  in  the  first  and 
second,  will  show  that  the  reasoning  of  that  article  is 
entirely  applicable  to  the  Hyperbola.  Hence, 

Theorem  XXXII,  —  The  diameters  which  pass  through 
the  feet  of  the  focal  perpendiculars  upon  any  tangent  of 
an  hyperbola,  are  parallel  to  the  corresponding  focal  radii 
of  contact. 

Corollary.  —  Hence,  also,  as  in  the  case  of  the  Ellipse, 
we  have  the  converse  theorem,  Diameters  parallel  to  the 
focal  radii  of  contact  meet 
the  tangent  at  the  feet  of  its 
focal  perpendiculars.  Con- 
sequently, after  finding  the 
foot  Q  of  the  focal  perpen- 
dicular, we  can  determine 
the  point  of  contact  T,  if 
we  wish  to  do  so,  by  sim- 
ply drawing  F'T  parallel  to  CQ. 

It  follows,  also,  that  the  distance  between  the  foot  of  the 
perpendicular  drawn  from  either  focus  to  a  tangent,  and 
tJte  foot  of  the  perpendicular  drawn  from  the  remaining 
focus  to  the  parallel  tangent,  is  constant,  and  equal  to  the 
length  of  the  transverse  axis. 

THE   NORMAL. 

497.  Equation  to  the  Normal.  —  From  the  equa- 
tion of  Art.  402,  by  changing  the  sign  of  62,  we  have 


_. 
-~ 


394 


ANALYTIC  GEOMETRY, 


498.  Let   PN  be   the   normal   to   an  hyperbola    at 


any  point  P"  and 
F'P  the  corresponding 
focal  radii.  The  equa- 
tions to  the  latter  (Art. 
95)  are 


Combining  the  equation 
to  the  normal  with  each 
of  these  in  succession,  we  get  (Art.  96) 


Hence,  FPN=l%Q°  —  F'PN=  QPN;  and  we  have 

Theorem  XXXIII.  —  The  normal  of  an  hyperbola  bisects 
the  external  angle  between  the  focal  radii  of  contact. 

Corollary  1.—  Comparing  Theorems   XXIV,   XXXIII 

of  the  Hyperbola  with  the  same  of  the  Ellipse,  we  at 
once  infer  :  If  an  ellipse  and  an  hyperbola  are  confocal, 
the  normal  of  the  one  is  the  tangent  of  the  other  at  their 
intersection. 

Corollary  2,  —  To  construct  a  normal  at  any  point  P 
of  the  curve,  we  draw  the  focal  radii  FP,  F'P,  produce 
one  of  them,  as  F'P,  until  PQ  =  FP,  and  join  QF:  then 
will  PN,  drawn  through  P  at  right  angles  to  QF,  be  the 
required  normal.  For  it  will  bisect  the  angle  FPQ,  accord- 
ing to  the  well-known  properties  of  the  isosceles  triangle. 

Corollary  3.  —  To  draw  a  normal  through  any  point  R 
on  the  conjugate  axis,  we  pass  a  circle  RF'R'F  through 
the  given  point  and  the  foci,  and  join  the  point  where 


CONSTRUCTION  OF  NORMAL. 


395 


this  circle  cuts  the  hyperbola  with  the  given  point  by 
the  line  RPN:  this  line  will  bisect  the  angle  FPQ, 
because  R  is  the  middle  point  of  the  arc  F'RF. 

It  is  important  to  notice,  however,  that  the  auxiliary 
circle  cuts  each  branch  of  the  curve  in  two  points,  as  P 
and  P',  and  that  only  one  of  these  (P,  in  the  diagram) 
answers  the  conditions  of  the  present  construction.  For 
the  line  joining  R  to  the  other,  as  RPr,  will  bisect  the 
internal  angle  between  the  focal  radii,  instead  of  the 
external.  We  thus  see  that  we  can  use  this  method  for 
drawing  a  tangent  from  any  point  in  the  conjugate  axis : 
a  statement  which  applies  to  the  Ellipse  also,  provided 
the  point  R  is  outside  of  the  curve. 


499.  Intercept  of 
the  Normal. — Making 
y  =  0  in  the  equation  of 
Art.  497,  we  obtain 


R 

K 


We  can  therefore,  as 
in  the  case  of  the  Ellipse 
(Art.  404),  construct  a 

normal  at  any  point  P  of  the  curve,  or  one  from  any 
point  N  of  the  transverse  axis. 

500.  By  an  argument  in  all  respects  similar  to  that 
of  Art.  405,  we  have  F'N  :  FN=  F'P  :  FP-,  that  is, 

Theorem  XXXIV. — The  normal  of  an  hyperbola  cuts  the 
distance  between  the  foci  in  segments  proportional  to  the 
adjacent  focal  radii  of  contact. 

501.  Length  of  the  Subnormal. — For  the  portion 
of  the  transverse  axis  included  between  the  foot  of  the 


396 


ANALYTIC  GEOMETRY. 


normal   and  that   of  the   ordinate  of  contact,  we   have 

MN=  ON      CM  =  e2x'  —  x'  =  (e2  —  1)  x'.     Hence, 

subnor  =  —  x'. 

a2 

502.  Comparing  the  results  of  Arts.  499  and   501, 
ON :  MN  =  c2  :  b2.     Or,  since  c2  =  a2  +  62,  we  have 

Theorem  XXXV. —  The  normal  of  an  hyperbola  cuts  the 
abscissa  of  contact  in  the  constant  ratio  (a2  +  b2)  :  b2. 

503.  Length  of  the  Normal. — Changing  the  sign 
of  b2  in  the  first  formula 

of  Art.  408,  and  then 
applying  the  formula  of 
Art.  473,  we  get 

W 


5O4.  Hence,  PNr.PR  =  b'2;  and  we  have 

Theorem  XXXVI. —  The  rectangle  under  the  segments 
formed  by  the  tivo  axes  upon  the  normal  is  equal  to  the 
square  on  the  semi-diameter  conjugate  to  the  point  of 
contact. 

Corollary.— Hence,  too,  (Art.  474)  PN.PR  =  ppr;  or, 

The  rectangle  under  the  segments  of  the  normal  is  equal 
to  the  rectangle  imder  the  focal  radii  of  contact. 

5O5o  Also  (Art.  489),  putting  Q  for  the  foot  of  the 
central  perpendicular  on  the  tangent  at  P,  CQ.PR  =  a2, 
and  OQ .  PN  =  b2.  That  is, 


SUPPLEMENTAL  CHORDS.  397 

Theorem  XXXVII. —  The  rectangle  under  the  normal 
and  the  central  perpendicular  upon  the  corresponding 
tangent  is  constant,  and  equal  to  the  square  on  the 
semi-axis  other  than  the  one  to  which  the  normal  is 

measured. 

SUPPLEMENTAL  AND  FOCAL  CHORDS. 

506.  Condition  that  Chords  of  an  Hyperbola 
be  Supplemental. — Let 

^,  (pf  denote  the  inclina- 
tions of  any  two  supple- 
mental chords  DP,  D'P. 
Then,  from  Art.  412,  by  /  , 

the  characteristic  change 
of  sign,  the  required  condition  will  be 

b2 
tan  <p  tan  ^'  —  — 2 . 

507.  Hence,  the  argument  of  Art.  413  applies  directly 
to  the  Hyperbola,  and  we  have 

Theorem  XXXVIII. — Diameters  of  an  hyperbola  which 
are  parallel  to  supplemental  chords  are  conjugate. 

Corollary  1. — To  construct  a  pair  of  conjugate  diam- 
eters at  a  given  inclination.  The  method  of  solving 
this  problem  in  the  Hyperbola  being  -identical  with 
that  given  for  the  Ellipse  in  the  first  corollary  to  Art. 
413,  we  do  not  consider  it  necessary  to  repeat  the 
details  here. 

Corollary  2. — To  construct  a  tangent  parallel  to  a  given 
right  line.  Let  LM\*Q  the  given  line.  Draw  any  diameter 
QR,  and  through  its  extremity  Q  pass  the  chord  QS 


398 


ANALYTIC  GEOMETRY. 


parallel  to  LM.  Form  the  supplemental  chord  SR 
and  its  parallel  di- 
ameter DP:  the  lat- 
ter, by  the  present 
theorem,  will  be  con- 
jugate to  that  drawn 
parallel  to  LM ;  and 
(Art.  485)  the  line 
PT,  drawn  through 
its  extremity  P,  and 
parallel  to  LM,  will  be  the  tangent  required. 

Corollary  3. —  To  construct  the  axes  in  the  empty  curve. 
Draw  any  two  parallel  chords,  bisect  them,  and  form  the 
corresponding  diameter,  say  QR.  On  the  latter,  describe 
a  semicircle  cutting  the  hyperbola  in  N.  Join  RN,  NQ, 
and  through  the  middle  point  of  QR  draw  A' A,  B'B  par- 
allel to  them  :  the  latter  will  be  the  axes,  by  the  same 
reasoning  as  that  used  in  Art.  413,  Cor.  3. 

5OS.  Focal  Chords. — The  properties  of  these  chords 
presented  in  Exs.  33 — 35,  p.  361,  are  as  true  for  the  Hy- 
perbola as  for  the  Ellipse.  The  reader  can  easily  con- 
vince himself  of  this  by  looking  over  his  solutions  of 
those  examples,  and  making  such  changes  in  the  formulae 
as  the  equation  to  the  Hyperbola  requires.  We  shall 
here  consider  only  that  single  property,  proved  for  the 
Ellipse  in  Art.  415,  which  serves  to  characterize  the 
parameter  of  the  curve. 

5O9.  For  the  length  of  any  focal  chord  in  an  hyper- 
bola, we  have,  by  changing  the  sign  of  b2  in  the  formula 
at  the  foot  of  p.  334, 

2  b2 

cho  —  -  .  -= — ^ ^  , 

a    e2cos2  v  —  1 


HYPERBOLA  REFERRED  TO  CONJUGATES.     399 

in  which  6  =  the  inclination  of  the  chord.  Hence 
(Art.  462),  putting  a'  -  -  the  semi-diameter  parallel  to 
the  chord, 


cho  = = 


a  2a 

That  is, 

Theorem  XXXIX. — Any  focal  chord  of  an  hyperbola 
is  a  third  proportional  to  the  transverse  axis  and  the 
diameter  parallel  to  the  chord. 

Remark. — The  latus  rectum  is  the  focal  chord  parallel 
to  the  conjugate  axis,  and  its  value  (Art.  454,  Cor.) 
exemplifies  this  theorem. 


ii.  THE  CURVE  REFERRED  TO  ANY  TWO  CONJUGATES. 

DIAMETRAL  PROPERTIES. 

51O.  Equation  to  the  Hyperbola,  referred  to 
any  two  Conjugate  Diameters.  —  The  equation  to 
the  primary  curve,  transformed  to  two  conjugates  whose 
respective  inclinations  are  6  and  0r,  is  found  by  simply 
changing  the  sign  of  b2  in  the  equation  at  the  middle 
of  p.  336.  It  is 

(a2sin20  —  62cos20)  x2  +  (a2sin20'  —  62cos20')  y2  =  — 


Hence,  by  changing  the  sign  of  the  constant  term,  the 
equation  to  the  conjugate  hyperbola,  referred  to  the  same 
pair  of  diameters,  is 


(a2sm2d  —  62cos20)  x2  -f  (a2sin20'  —  62cos20')  y1  =  a2b2. 

Now  let  a',  b'  denote  the  lengths  of  the  semi-diameters 
An.  Ge.  37. 


400  ANALYTIC  GEOMETRY. 

of  reference  :  we  shall  get,  by  making  y  =  0  in  the  first 
of  these  equations,  and  x  =  0  in  the  second, 

a2sin20  —  62cos2#  =  —  a^~  ,      a2sin20'  —  62cos20'  =  —  . 
a'2  b'2 

Substituting  in  the  first  equation  above,  we  obtain 

x*_        tf_ 
a!'2        b'2  ~ 

Corollary  1.  —  The  transformed  equation  to  the  conju- 
gate curve  is  therefore 


.    — 

a12  ~     b'2 

Moreover  (since  a'2—b'2  =  a2  —  62),  in  the  Equilateral 
Hyperbola  we  have  b'  =  a'  :  hence,  the  equations  to 
that  curve  and  its  conjugate,  referred  to  any  two  conju- 
gate diameters,  are 

x2  —  y2=±a'2. 

Corollary  2.  —  The  new  equation  to  the  Hyperbola 
differs  from  the  analogous  equation  to  the  Ellipse  (Art. 
417),  only  in  the  sign  of  bf2.  Hence,  Any  function  o/b' 
thai  expresses  a  property  of  the  Ellipse,  will  be  converted 
into  one  expressing  a  corresponding  property  of  the  Hyper- 
bola by  merely  replacing  its  b'  by  V  V  —  1. 

511.  The  remarks  of  Art.  418  evidently  apply  to  the 
equations 


- 

a2        b2  ~ 

Hence,  we  have  the  following  extensions  of  Theorems 
VI,  VII: 


DIAMETRAL  PROPERTIES.  401 

Theorem  XL.  —  The  squares  on  the  ordinates  to  any 
diameter  of  an  hyperbola  are  proportional  to  the  rectangles 
under  the  corresponding  segments  of  the  diameter. 

Theorem  XLI.  —  The  square  on  any  diameter  of  an  hy- 
perbola is  to  the  square  on  its  conjugate,  as  the  rectangle 
under  any  two  segments  of  the  diameter  is  to  the  square 
on  the  corresponding  ordinate. 

512.  Writing  the  equation  of  Art.  510  in  the  form 


and  comparing  it  with  that  of  the  Equilateral  Hyperbola, 
namely,  with 


we  get  yh  :  yr  =  bf  :  a!  .     That  is,   as  the  extension  of 
Theorem  VIII, 

Theorem  XLII.  —  The  ordinate  to  any  diameter  of  an 
hyperbola  is  to  the  corresponding  ordinate  of  its  equi- 
lateral, as  the  conjugate  semi-diameter  is  to  the  semi- 
diameter. 

Remark  —  We  may  take  the  corresponding  ordinate  of  the  equi- 
lateral as  signifying  either  the  oblique  ordinate  of  the  equilateral 
described  upon  the  same  transverse  axis  as  the  given  hyperbola,  or 
the  rectangular  ordinate  of  the  equilateral  described  upon  the  diam- 
eter selected  for  the  axis  of  x.  For  the  equation  x2  —  y1  =  a'2  will 
denote  either  of  these  equilaterals,  according  as  it  is  supposed  to 
refer  to  oblique  or  rectangular  axes.  Only  we  must  understand 
that,  in  either  interpretation,  the  corresponding  ordinates  are  those 
which  have  a  common  abscissa. 

It  is  evident,  also,  that  the  ratio  between  the  corresponding 
ordinates  of  the  hyperbola  and  the  circle  x2  -f-  y1  =  a'2,  described 
on  any  diameter  of  the  curve,  is  imaginary.  Hence,  with  respect 
to  this  circle,  there  is  a  defect  in  the  analogy  between  the  Ellipse 


402  ANALYTIC  GEOMETRY. 

and  the  Hyperbola :   a  defect  that  will  be  supplied,  however,  as 
soon  as  we  develop  the  conception  of  the  subtangent  to  any  diameter. 

513.  We  leave  the  student  to  show,  by  interpreting 
the  equation 

7/2 


that,  with  reference  to  any  diameter,  the  Hyperbola 
consists  of  two  infinite  branches,  extending  in  opposite 
directions,  and  both  symmetric  to  the  diameter. 

CONJUGATE    PROPERTIES    OP    THE    TANGENT. 

514.  Equation  to  the  Tangent,  referred  to  any 
two  Conjugate  Diameters. — By  changing  the  sign 
of  bf2  (Art.  510,  Cor.  2)  in  the  equation  of  Art.  421, 
the  equation  now  sought  is  seen  to  be 


515.  Intercept  of  the  Tangent  on  any  Diam- 
eter.— Making    y  =  0    in 
the  equation  just  found,  we 

get,    for    the    intercept    in  .  ,,    _._, 

question, 


x 

Hence,  as  the  extension  of  Art.  487, 

Theorem  XLIII, —  The  intercept  cut  off  by  a  tangent 
upon  any  diameter  of  an  hyperbola  is  a  third  propor- 
tional to  the  abscissa  of  contact  and  the  semi-diameter. 

Corollary, — To   construct   a   tangent  from   any  given 


INTERCEPTS  OF  TANGENTS.  403 

point.  The  method  of  the  corollary  to  Art.  422  applies 
directly  to  the  Hyperbola,  and  the  student  may  interpret 
the  statements  there  made,  as  referring  to  the  present 
diagram  letter  by  letter. 

516.  The  properties  of  tangential  intercepts,  proved 
in  Art.  423  with  respect  to  the  Ellipse,  are  also  true  of 
the  Hyperbola.  We  shall 
merely  restate  them  here, 
leaving  the  reader  to  make 
such  simple  modifications  of 
the  analyses  in  I,  II,  III 
of  the  article  mentioned,  as 
may  be  necessary  to  establish 
them.  To  aid  him  in  this,  the  parts  of  the  annexed  dia- 
gram are  lettered  identically  with  the  corresponding  parts 
of  that  in  Art.  423. 

I.  Theorem  XLIV.  —  The  rectangle  under  the  intercepts  cut  off 
upon  two  fixed  parallel  tangents  by  any  variable  tangent  of  an  hyper- 
bola is  constant,  and  equal  to  the  semi-diameter  parallel  to  the  two 
tangents. 

II.  Theorem  XLV. — The  rectangle  under  the  intercepts  cut  off 
upon  any  variable  tangent  of  an  hyperbola  by  two  fixed  parallel 
tangents  is  variable,  being  equal  to  the  square  on  the  semi-diameter 
parallel  to  the  tangent. 

III.  Theorem  XLVI. —  The  rectangle  under  the  intercepts  cut  off 
upon  any  variable  tangent  of  an  hyperbola  by  two  conjugate  diameters 
is  equal  to  the  square  on  the  semi-diameter  parallel  to  the  tangent. 

Corollary  1. — By  the  same  reasoning  as  in  the  first  corollary  to 
III  of  Art.  423,  we  have:  Diameters  drawn  through  the  intersections 
of  any  tangent  with  two  parallel  tangents  are  conjugate. 

Corollary  2. — The  problem,  Given  two  conjugate  diameters  of  an 
hyperbola  in  position  and  magnitude,  to  construct  the  axes,  is  solved 
by  the  same  process  as  the  corresponding  one  on  p.  342;  excepting 
that  the  point  P  must  be  taken  on  the  side  of  D  next  to  C,  instead 
of  on  the  side  remote  from  it. 


404  ANALYTIC  GEOMETRY. 

517.  Subtangent   to    any    Diameter.  —  For    the 

length   of   this,    we    have    MT'=CM—CTf.     Hence, 
putting  xf  =  CM,  and  sub- 
stituting  the   value   of  CTr 
from  Art.  515, 

.       xn  —  a'2 
subtan  = 


x' 
That  is,  since  x'  +  a'  =  LM, 


tangent  to  any  diameter  of  an  hyperbola  is  a  fourth  pro- 
portional to  the  abscissa  of  contact  and  the  corresponding 
segments  of  the  diameter. 

Corollary. — If  we  compare  this  value  of  the  general 
subtangent  with  that  of  the  subtangent  of  the  curve 
(Art.  488),  we  see  at  once  that  the  argument  used  in 
Art.  488,  Cor.  1,  with  respect  to  the  Hyperbola  and  its 
inscribed  circle,  applies  to  the  curve  and  the  circle  de- 
scribed upon  any  of  its  diameters.  Hence,  If  through 
the  head  and  foot  of  an  ordinate  to  any  diameter  of  an 
hyperbola  tangents  be  drawn  to  the  curve  and  to  the  circle 
described  upon  the  diameter,  they  will  have  a  common 
subtangent. 

In  other  words,  if  Q  is  the  point  in  which  a  rectangular 
ordinate  drawn  through  the  foot  of  a  tangent  to  the  hy- 
perbola pierces  the  circle  mentioned,  the  tangent  to  this 
circle  at  Q  passes  through  M,  the  foot  of  the  ordinate  of 
contact  for  the  tangent  to  the  hyperbola.  The  defect 
noticed  in  the  Remark  under  Art.  512,  is  therefore 
supplied;  and  we  may  employ  the  circle  in  question,  to 
solve  the  following  problem  : 

To  draw  a  tangent  to  an  hyperbola  from  any  given  point. 
Let  T'  be  the  given  point.  Draw  the  diameter  DT'L, 
and  form  the  corresponding  circle  C-  DQL.  At  the  given 


PARAMETER  OF  THE  HYPERBOLA.  405 

point,  set  up  T'Q  a  rectangular  ordinate  to  this  circle, 
and  through  its  extremity  Q  draw  the  tangent  QM. 
Then,  through  the  foot  M  of  this  tangent,  pass  MP 
parallel  to  the  diameter  conjugate  to  DL  :  the  point  P 
in  which  this  parallel  cuts  the  hyperbola,  will  be  the 
point  of  contact  of  the  required  tangent,  which  may  be 
obtained  by  joining  T'P. 

Remark,  —  To  form  a  tangent  at  any  point  P  of  the 
curve,  we  draw  the  ordinate  PM,  and,  through  its  foot, 
the  circular  tangent  MQ.  Then,  if  QT'  be  drawn  at 
right  angles  to  the  diameter  DL,  Tf  will  be  the  foot 
of  the  required  tangent. 

518.  By  the  same  reasoning  as  in  Art.  425,  we  get 

Theorem  XL  VII.  —  The  rectangle  under  the  subtangent 
and  the  abscissa  of  contact  is  to  the  square  on  the  ordinate 
of  contact,  as  the  square  on  the  corresponding  diameter  is 
to  the  square  on  its  conjugate. 

519.  Changing  the  sign  of  bf2  in  the  equations  of  Art. 
426,  and  then  taking  the  steps  indicated  there,  we  obtain 

Theorem  XL  VIII.  —  Tangents  at  the  extremities  of  any 
chord  of  an  hyperbola  meet  on  the  diameter  which  bisects 
that  chord. 

PARAMETERS. 

520.  Definitions.  —  The  Parameter  of  an  hyperbola, 
with  respect  to  any  diameter,  like  the  parameter  of  an 
ellipse,  is  a  third  proportional  to  the  diameter  and  its 
conjugate.     Thus, 


parameter  =  -r—  f-  = 
*la! 


406  ANALYTIC  GEOMETRY. 

The  parameter  with  respect  to  the  transverse  axis,  is 
called  the  principal  parameter;  or,  the  parameter  of  the 
curve.  We  shall  denote  its  length  by  4p. 

521.  For  the  value  of  the  parameter  of  the  Hyper- 
bola, we  accordingly  have 


Thus  (Art.  454,  Cor.)  the  principal  parameter  is  identical 
with  the  latus  rectum,  and  may  therefore  be  described  as 
the  double  ordinate  to  the  transverse  axis,  drawn  through 
the  focus. 


In  Art.  509,  we  proved  that  the  focal  double 
ordinate  parallel  to  any  diameter  is  a  third  proportional 
to  the  transverse  axis  and  the  diameter.  Now  (Art.  463) 
the  transverse  axis  is  less  than  any  other  diameter  —  less, 
therefore,  than  the  diameter  conjugate  to  that  of  which 
the  focal  chord  is  a  parallel,  unless  the  chord  is  the  latus 
rectum.  Hence, 

Theorem  XLIX, — No  parameter  of  an  hyperbola,  except 
the  principal,  is  equal  in  value  to  the  corresponding  focal 
double  ordinate. 

POLE   AND    POLAR. 

52Bo  We  now  proceed  to  develop  the  polar  relation 
as  a  property  of  the  Hyperbola;  and  shall  follow  the 
steps  already  twice  taken,  in  connection  with  the  Circle 
and  the  Ellipse. 

524.  Chord  of  Contact  in  the  Hyperbola. — Let 

x'y'  be  the  fixed  point  from  which  the  two  tangents  that 


POLAR  IN  THE  HYPERBOLA.  407 

determine  the  chord  are  drawn.  Then,  by  merely  chang- 
ing the  sign  of//2  in  the  equation  of  Art.  481,  the  equation 
to  the  hyperbolic  chord  of  contact  will  be 


__  -, 
a'2     ~   b'2~ 


525.  Locus  of  the  Intersection  of  Tangents  to  the 
Hyperbola.  —  Let  x'y'  denote  the  fixed  point  through 
which  the  chord  of  contact  belonging  to  any  two  of  the 
intersecting  tangents  is  drawn,  and  change  the  sign  of 
b12  in  the  equation  of  Art.  432  :  the  equation  to  the  locus 
now  considered  will  then  be 


b'2 

526.  Tangent  and  Chord  of  Contact  taken  up 
into  the  wider  conception  of  the  Polar.  —  From  the 
identity  in  the  form  of  the  last  two  equations  with  the 
form  of  the  equation  to  the  tangent,  we  see  that,  in  the 
Hyperbola  also,  the  law  which  connects  the  tangent  with 
its  point  of  contact,  and  the  chord  of  contact  with  the 
point  from  which  its  determining  tangents  are  drawn,  is 
the  same  that  connects  the  locus  of  the  intersection  of 
tangents  drawn  at  the  extremities  of  chords  passing 
through  a  fixed  point,  with  that  point. 

In  short,  the  three  right  lines  represented  by  these 
equations  are  only  different  expressions  of  the  same  formal 
law  :  a  law,  moreover,  of  which  the  locus  mentioned  is  the 
generic  expression.  For,  in  the  case  of  the  tangent,  the 
point  x'y'  is  restricted  to  being  on  the  curve  ;  and,  in  that 
of  the  chord  of  contact,  to  being  within  ;  while,  in  that  of 
the  locus,  it  is  unrestricted:  so  that  the  tangent  and  the 
chord  of  contact  are  cases  of  the  locus,  due  to  bringing 


408  ANALYTIC   GEOMETRY. 

the  point  x'y'  upon  the  curve  or  within  it.  Moreover,  the 
formal  law  which  connects  the  locus  with  the  fixed  point 
is  the  law  of  polar  reciprocity.  For,  by  its  equation,  the 
locus  is  a  right  line  ;  and,  if  we  suppose  the  point  x'y'  to 
be  any  point  on  a  given  right  line,  the  co-efficients  of  the 
equation  in  Art.  524  will  be  connected  by  the  relation 
Ax'  +  Byf+  0=0',  whence  (Art.  117)  we  have  the 
twofold  theorem: 

I.  If  from   a  fixed   point    chords   be    drawn   to   any 
hyperbola,  and  tangents  to  the  curve   be  formed  at   the 
extremities  of  each  chord,  the  intersections  of  the  several 
pairs  of  tangents  will  lie  on  one  right  line. 

II.  If  from  different  points  lying  on  one  right  line 
pairs  of  tangents  be  drawn  to  any  hyperbola,  their  several 
chords  of  contact  ivill  meet  in  one  point. 

The  Hyperbola,  then,  imparts  to  every  point  in  its 
plane  the  power  of  determining  a  right  line  ;  and  recip- 
rocally. 


Equation  to  the  Polar  with  respect  to  an 
Hyperbola.  —  From  the  conclusions  now  reached,  this 
equation,  referred  to  any  two  conjugate  diameters,  must 
be 

^_?/V_-|. 
a'2     '    b'*~ 

or,  referred  to  the  axes  of  the  curve, 

2/3        /#_-, 
a*   ~  "  #  = 

x'y'  being  the  point  to  which  the  polar  corresponds. 

<52S.  Definitions.  —  The  Polar  of  any  point,  with  re- 
spect to  an  hyperbola,  is  the  right  line  which  forms  the 
locus  of  the  intersection  of  the  two  tangents  drawn  at  the 
extremities  of  any  chord  passing  through  the  point. 


POLAR  IN  THE  HYPERBOLA. 


409 


The  Pole  of  any  right  line,  with  respect  to  an  hyper- 
bola, is  the  point  in  which  all  the  chords  of  contact  corre- 
sponding to  different  points  on  the  line  intersect. 

Hence  the  following 
constructions :  —  When 
the  pole  P  is  given, 
draw  through  it  any  two 
chords  T'T,  S'S,  and 
form  the  corresponding 
pairs  of  tangents,  T'L 
and  TL,  S'M  and  Slf : 
the  line  LM,  which  joins 

the  intersection  of  the  first  pair  to  that  of  the  second, 
will  be  the  polar  of  P.  When  the  polar  is  given,  take 
upon  it  any  two  points,  as  L  and  M,  and  draw  from 
each  a  pair  of  tangents,  LT  and  LT',  MS  and  MS': 
the  point  P,  in  which  the  corresponding  chords  of 
contact  T'T,  S'S  intersect,  will  be  the  pole  of  LM. 

This  construction  is  applicable  in  all  cases;  and,  when 
the  pole  is  without  the  curve,  as  at  Q,  it  must  be  used. 
But  if  the  pole  is  within  the  curve,  as  at  P,  the  polar 
LM  may  be  obtained  by  drawing  the  chord  of  contact 
of  the  two  tangents  from  P;  and  if  it  is  on  the  curve, 
as  at  T,  the  polar  is  the  corresponding  tangent  LT. 

529.  Direction   of  the   Polar. — By   changing   the 
sign  of  bf2  in  the  equations  of  Art.  436,  and  then  using 
the  principle  of  inference  employed  there,  we  obtain 

Theorem  L. —  The  polar  of  any  point,  ivitli  respect  to 
an  hyperbola,  is  parallel  to  the  diameter  conjugate  to  that 
ivliicli  passes  through  the  point. 

530.  Polars  of  Special  Points. — A  comparison  of 
the  equation  to  the  polar  in  an  hyperbola  with  its  equa- 
tion as  related  to  the  Circle  (Art.  323),  will  show  that 


410  ANALYTIC  GEOMETRY. 

the  general  properties  proved  of  polars  in  Art.  324  are 
true  for  the  Hyperbola.  We  therefore  pass  at  once  to 
those  special  properties  which  characterize  the  polars  of 
certain  particular  points. 

Applying  the  processes  of  Art.  437  to  the  equation  of 
Art.  527,  we  get 

I.   The  polar  of  the  center  is  a  right  line  at  infinity. 

II.   The  polar  of  any  point  on  a  diameter  is  a  right 

line  parallel  to  the  conjugate  diameter,  and  its  distance 

from  the  center  is  a  third  proportional  to  the  distance  of 

the  point,  and  the  length  of  the  semi- diameter. 

III.  The  polar  of  any  point  on  the  transverse  axis  is 
the  perpendicular  whose  distance  from  the  center  is  a 
third  proportional  to  the  distance  of  the  point  and  the 
length  of  the  semi-axis. 

Corollary. — From  II  it  follows,  that  the  construction 
for  the  polar,  given  under  Art.  437  with  respect  to  the 
Ellipse,  is  entirely  applicable  to  the  Hyperbola. 

531.  Polar  of  the  Focus. — The  equation  to  this  is 
found  by  putting  (±  ae,  0)  for  x'y'  in  the  second  equa- 
tion of  Art.  527,  and  is  therefore 


Hence,  The  polar  of  either  focus  in  an  hyperbola  is  the 
perpendicular  which  cuts  the  transverse  axis  at  a  distance 
from  the  center  equal  to  a  :  e,  measured  on  the  same  side 
as  the  focus. 

Remark. — Since  the  e  of  the  Hyperbola  is  greater  than 
unity,  the  distance  of  the  focal  polar  from  the  center  is 
in  that  curve  less  than  a.  In  the  Ellipse,  on  the  contrary, 
this  distance  is  greater  than  a,  because  the  e  of  that  curve 


POLAR  OF  THE  FOCUS. 


411 


is  less  than  unity.  Hence,  in  the  Ellipse,  the  polar  of  the 
focus  is  without  the  curve ;  but,  in  the  Hyperbola,  it  is 
situated  within. 

532.  The  distance  of  any  point  P  of  an  hyperbola 
from  either  focal  polar,  for  instance 
from  DR,  is  obviously  equal  to  the 
abscissa  of  the  point,  diminished  by 
the  distance  of  the  polar  from  the 
center.  That  is, 

PD=x--  =  ex  ~  a  . 

e  e 

But  (Art.  457)  ex— a  =FP.    Therefore, 
FP 


PD 


=  e. 


In  other  words,  the  property  of  Art.  439  re-appears, 
and  we  have 

Theorem  LI. —  The  distance  of  any  point  on  an  hyper- 
bola from  the  focus  is  in  a  constant  ratio  to  its  distance 
from  the  polar  of  the  focus,  the  ratio  being  equal  to  the 
eccentricity  of  the  curve. 

Corollary  1. — We  may  therefore  describe  an  hyperbola 
by  a  continuous  motion,  as  follows : 

Take  any  point  F,  and  any  fixed  right  line  DR.  Against  the 
latter,  fasten  a  ruler  DD/1  and  place  a  second 
ruler  NQL  (right-angled  at  L)  so  that  its 
edge  LN  may  move  freely  along  DD'.  At 
F  fasten  one  end  of  a  thread  equal  in  length 
to  the  edge  NQ  of  this  last  ruler,  to  whose 
extremity  Q  the  other  end  must  be  attached. 
Then,  with  the  point  P  of  a  pencil,  stretch 
this  thread  against  the  edge  NQ,  and  move 
the  pencil  so  that  the  thread  shall  be  kept 
stretched  while  the  ruler  NQL  slides  along 
DD/ :  the  path  of  P  will  be  an  hyperbola.  For.  by  the  conditions 


412  ANALYTIC  GEOMETRY. 

named,  FP  must  equal  PN  in  every  position  of  the  pencil:  whence 
FP  :  PD  —  NQ  :  QL.  That  is,  since  DR  must  be  the  polar  of 
F  with  respect  to  some  hyperbola,  the  focal  distance  of  P  is  in  a 
constant  ratio  to  its  distance  from  the  focal  polar. 

This  construction  derives  interest  from  a  comparison 
with  that  of  the  Ellipse  in  Art.  439,  Cor.  1.  It  will  be 
seen  that  the  essential  principle  is  the  same  in  both, 
namely,  the  use  of  the  parts  of  a  right  triangle  to  de- 
termine a  constant  ratio  between  the  distances  of  points 
from  a  fixed  point  and  a  fixed  right  line.  It  is  notice- 
able, that,  in  the  Ellipse,  this  constant  ratio  is  that  of 
the  base  to  the  hypotenuse  ;  while,  in  the  Hyperbola,  it 
is  that  of  the  hypotenuse  to  the  base  ;  thus  illustrating 
the  inverse  relation  existing  between  the  two  curves. 

Corollary  2.  —  In  the  construction  just  explained,  the 
polar  of  the  focus  is  used  as  the  directing  line  of  the 
motion  which  generates  the  curve.  For  this  reason,  it 
is  called  the  directrix  of  the  corresponding  hyperbola. 

Corollary  3.  —  In  the  light  of  the  present  theorem,  we 
may  interpret  the  name  hyperbola  as  denoting  the  conic 
in  which  the  constant  ratio  between  the  focal  and  polar 
distances  exceeds  unity. 


Focal  Angle  subtended  by  any  Tangent.  —  By  exam- 
ining the  investigation  conducted  in  Art.  440,  the  student  will  see 
that  it  is  applicable  to  the  Hyperbola,  with  the  single  exception  of 
a  change  in  the  sign  of  the  final  result.  Hence,  if  p  =  the  focal 
distance  of  any  given  point  from  which  a  tangent  is  drawn  to  an 
hyperbola,  and  x  =  the  abscissa  of  the  point,  the  angle  9  which  the 
portion  of  the  tangent  intercepted  between  the  given  point  and  the 
point  of  contact  subtends  at  the  focus,  will  be  determined  by  the 
formula 

ex—  a 

cos  0  =  -  • 
fi 

534.    This  expression,  being  independent  of  the  point  of  con- 
tact a/?/,  would  seem  to  indicate  that  both  of  the  tangents  that  can 


HYPERBOLA  REFERRED  TO  ITS  FOCI.         413 

be  drawn  from  a  given  point  to  the  curve  subtend  the  same  focal 

angle.     It  is  found,  however,  as  in  the  case  presented  in  the  dia- 

gram, that  when  the  given  point  P  is  taken  under  such  conditions 

(Art.  484,  Cor.  2)  as  fix  the  two 

points  of  contact  T  and  T/  on 

opposite  branches  of  the  curve, 

the  angles  PFT,  PFT/  are  not 

equal,    but    supplemental.      But, 

whether  they  be  the  one  or  the 

other,  the  line  FP  must  bisect 

the  whole  angle  T'FT  subtended 

by  the  chord  of  contact,  either  internally  or  externally.     Hence, 

Theorem  LIT.  —  The  right  line  that  joins  the  focus  to  the  pole  of  any 
chord,  bisects  the  focal  angle  which  the  chord  subtends. 

Corollary.  —  The  angle  subtended  by  a  focal  chord  being  180°, 
we  have,  as  a  special  case  of  the  preceding:  The  line  that  joins 
the  focus  to  the  pole  of  any  focal  chord  is  perpendicular  to  the  chord. 

in.  THE  CURVE  REFERRED  TO  ITS  Foci. 

535.  In  the  polar  equations  of  Art.  172  and  the 
subjoined  Remark,  namely,  in 

_  a  (1  —  e2)  a  (e2  —  1) 


we  now  know  that  the  constant  a  is  the  transverse  semi- 
axis  of  the  corresponding  hyperbola,  and  the  constant  e 
its  eccentricity. 

Replacing,  then,  e2  —  1  by  its  value  (Art.  171)  b2  :  a2, 
we  may  write  these  equations 

b2  1  b2  1 


P —        ~  '  -i  ~~  a  '      P  —  „ 


a    1  —  ecosti  a     1  —  ecosO. 

But  (Art.  521)  b2 :  a  is  half  the  parameter  of  the  curve. 


414  ANALYTIC  GEOMETRY. 

the  upper  sign  corresponding  to   the   right-hand  focus, 
and  the  lower  to  the  left-hand. 

536.  Polar    Equation    to    the    Tangent.  —  By    an 

analysis  exactly  similar  to  that  in  Art.  444,  we  find  this 
to  be 


cos  (0  —  ti')  —  e  cos  0 

Corollary.  —  The  equation  to  the  diameter  conjugate  to 
x'y'  (Art.  469)  differs  from  that  of  the  tangent  at  x'y' 
only  in  having  0  for  its  constant  term.  Hence,  as  in 
the  corollary  to  Art.  444, 

ae  (cos  6'  —  e) 


cos  (6  —  0')  —  e  cos  6 

is   the  polar  equation  to  the  diameter  conjugate  to  that 
which  passes  through  p'6'. 

iv.  THE  CURVE  REFERRED  TO  ITS  ASYMPTOTES. 

537.  Hitherto,    the    properties    established    for    the 
Hyperbola  have  had  a  fixed  relation,  either  of  identity 
or  of  antithesis,  to  those  of  the  Ellipse.     We  now  come, 
however,  to  a  series  of  properties  peculiar  to  the  Hyper- 
bola, arising  from  the  presence  of  the  two  lines  which 
we  have  named  the  self-conjugate  diameters.    We  might 
proceed  at  once  to  transform  the  equation  of  Art.  167 
to   these   diameters    as   axes   of  reference ;    but,   before 
doing  so,  let  us  subject  the   self-conjugates  themselves 
to  a  more  minute  examination. 

538.  Definition. — An  Asymptote  of  any  curve  is  a 
line  which  continually  approaches  the  curve,  but  meets 
it  only  at  infinity. 


ASYMPTOTES  OF  THE  HYPERBOLA.  415 

Asymptotes  are  either  curvilinear  or  rectilinear.  The 
term  asymptote  is  derived  from  the  Greek  a  privative, 
and  0uu7ii~T£gv,  to  coincide,  and  may  be  taken  as  signi- 
fying that  the  line  to  which  it  is  applied  never  meets  the 
curve  which  it  forever  approaches. 

53O.  We  have  already  once  or  twice  spoken  of  the 
self-conjugates  of  the  Hyperbola  as  its  asymptotes.  We 
now  proceed  to  show  that  they  are  such.  We  have 
proved  (Arts.  461,  Cor.  1;  480)  that  they  meet  the 
curve  only  at  infinity  :  it  remains  to  show  that  they 
draw  nearer  and  nearer  to  the  curve  the  farther  they 
recede  from  the  center. 

Let  CM  be  any  common  abscissa  of  an  hyperbola  and 
its  self-conjugate  diameter  CL.  The  equations  to  CL 
and  the  curve  being  respectively 


we  get,  for  the  difference  between 
any  two  corresponding  ordinates, 


T>n        t>  (  ~TI\  ^ 

PQ=     („-</*•- a)  =  - 


Hence,  as  x  increases,  PQ  diminishes ;  so  that,  if  we 
suppose  x  to  be  increased  without  limit,  or  the  point  P 
of  the  curve  to  recede  to  an  infinite  distance  from  the 
origin,  PQ  will  converge  to  the  limit  0.  Now  the 
distance  of  any  point  P  of  the  curve  from  the  self- 
conjugate  CL  is  equal  to  PQsmPQC:  therefore,  as 
the  angle  PQC  is  the  same  for  every  position  of  P, 
this  distance  diminishes  continually  as  P  recedes  from 
the  center ;  or,  we  have 
An.  Ge.  38. 


416  ANALYTIC  GEOMETRY. 

Theorem  LIII. —  The  self-conjugate  diameters  of  an 
hyperbola  are  asymptotes  of  the  curve. 

Remark. — We  have  inferred  this  theorem  with  respect  to  L' 'R 
as  well  as  LR',  although  the  preceding  investigation  is  conducted 
in  terms  of  LR/  only.  But  it  is  manifest  that  a  similar  analysis 
applies  to  L'R;  and  it  can  be  shown,  in  like  manner,  that  LR' 
and  ISR  are  asymptotes  of  the  conjugate  hyperbola.  We  leave  the 
proof  of  this,  however,  as  an  exercise  for  the  student. 

51O.  Angle  between  the  Asymptotes. — From  Art. 

479,  we  have  tan  LCM=  b  :  a,  and  tan  L'CM=  -  b  :  a ; 
hence  LCM=  l%Q°—LfCM=  RCM.  If,  then,  we  put 
<p  =  the  required  angle  LCR,  and  6  =  LCM,  we  shall 
get  <p  =  20 ;  or,  since  tan  6  =  b  :  a,  and  therefore  cos  0  = 

a  :  <?, 

<p  =  2  sec"1  e. 

Hence,  if  the  eccentricity  of  an  hyperbola  is  given, 
the  inclination  of  its  asymptotes  is  also  given  ;  for  it  is 
double  the  angle  whose  secant  is  the  eccentricity.  Con- 
versely, when  the  inclination  of  the  asymptotes  is  known, 
the  eccentricity  is  found  by  taking  the  secant  of  half  the 
inclination. 

Thus,  in  the  case  of  an  equilateral  hyperbola,  whose 
eccentricity  (Art.  456,  Cor.)  —  1/2,  we  have 

<p  =  2  sec-1 1/2  =  90°: 

which  agrees  with  the  property  by  which  (Art.  177,  Cor.) 
we  originally  distinguished  this  curve. 

541.  Equations  to  the  Asymptotes.  —  These  are 
respectively  (Art.  479)  y  =  (b  :  a)  x,  y  =  —  (b  :  d)  x. 
Or  we  may  write  them 

x    y       x  -L  y 

~ 


ASYMPTOTES  OF  THE  HYPERBOLA.  417 

Hence,  (Art.  124,)  the  equation  to  both  asymptotes  is 


F 


542.  Let  CD,  CD'  be  any  two  conjugate  semi-diam- 
eters.    Then,  from   the    fact    that 
the    equation    to    the    Hyperbola,  L,  ^ 
when  referred  to  these,  is  identical 
in  form  with  its  equation  as  referred 
to  the  axes,  we  may  at  once  infer 
that 


are  the  equations  to  the  asymptotes,  referred  to  any  pair 
of  conjugates. 

Now  the  first  of  these  lines  (Art.  95,  Cor.  2)  passes 
through  the  point  a'b1 ',  that  is,  through  the  vertex  of  the 
parallelogram  formed  on  CD  and  CD ';  while  the  second 
(Art.  98,  Cor.)  is  parallel  to  the  line 

x         y        1 
a7  +  F=      ' 

that  is,  to  the  diagonal  D'D  of  the  same  parallelogram. 
Hence,  the  asymptotes  have  the  same  direction  as  the 
diagonals  of  this  parallelogram  ;  or,  extending  the  prop- 
erty to  the  figure  of  which  this  parallelogram  is  the 
fourth  part,  we  get 

Theorem  LIV. —  The  asymptotes  are  the  diagonals  of 
every  parallelogram  formed  on  a  pair  of  conjugate  diam- 
eters. 

Corollary. — If,  then,  we  have  any  two  conjugate  diam- 
eters given,  we  can  find  the  asymptotes ;  and,  conversely, 


418 


ANALYTIC  GEOMETRY. 


given  the  asymptotes  and  any  diameter  CD,  we  can  find 
its  conjugate  by  drawing  DO  parallel  to  CR,  and  pro- 
ducing it  till  OD'  =  OD,  when  D'  will  be  the  extremity 
of  the  conjugate  sought. 

543.  From  the  equation  to  the  tangent  (Art.  481) 
we  get 


or,  after  substituting  for  y'  from 
the  equation  to  the  curve,  and 
factoring, 

6  1 


Supposing,  then,  that  x'  and  y'  are  increased  without 
limit,  or  that  the  point  of  contact  P  recedes  to  an  infinite 
distance  from  the  origin,  the  limiting  form  to  which  this 
equation  tends  is 

b 


But  this  (Art.  541)  is  the  equation  to  CL  ;  and  a  like 
result  can  be  readily  obtained  with  respect  to  the  other 
asymptote.  Hence, 

Theorem  LV.  —  The  asymptotes  are  the  limits  to  which 
the  tangents  of  an  hyperbola  converge  as  the  point  of 
contact  recedes  toivard  infinity. 

Kemark  —  We  might  therefore  define  the  asymptotes  ns  the  right 
lines  which  meet  the  Jtyperbola  in  two  consecutive  points  at  infinity. 

544.  Accordingly,  by  Theorem  XXIX  (Art.  493),  the 

product  of  the  focal  perpendiculars  upon  an  asymptote 


ASYMPTOTES  OF  THE  HYPERBOLA. 


419 


must  be  equal  to  b2.  But,  since  the  asymptote  passes 
through  the  center,  these  focal  perpendiculars  must  be 
equal  to  each  other,  and  therefore  each  equal  to  b. 
That  is, 

Theorem  LVI. —  The  perpendicular  from  either  focus  to 
an  asymptote  is  equal  to  the  conjugate  semi-axis. 

545.  Let  FP  be  the  focal  dis- 
tance of  any  point  on  an  hyperbola, 
and  PD  its  distance  from  the  direc- 
trix DR.  By  Art.  532,  FP=e.PD. 
But  (Art.  540),  e  =  secLCF.  Hence, 
FP=PDsecLCF=PDcosecPRD, 
if  PR  be  drawn  parallel  to  the  asymp- 
tote CL.  That  is  (Trig.,  859),  FP  = 
PR-,  or,  we  have 

Theorem  LVII. —  The  focal  distance  of  any  point  on  an 
hyperbola  is  equal  to  its  distance  from  the  adjacent  direc- 
trix, measured  on  a  parallel  to  either  asymptote. 


Corollary — We  here  find  a  new  reason 
for  the  method  of  generating  an  hyperbola, 
given  in  the  first  corollary  to  Art.  532. 
For,  by  the  requirements  of  the  method, 
FP=  PR  =  e.PD.  Now,  by  the  diagram, 
PR  =  PD  sec  RPD.  Hence,  the  method 
makes  sec-RPD  =  «;  that  is  (Art.  540),  it 
makes  the  angle  RPD  equal  to  the  inclina- 
tion of  the  asymptote,  and  PR  therefore  par- 
allel to  that  line. 


546.  Equation  to  the  Hyperbola,  referred  to  its 
Asymptotes. — The  equation  to  the  Hyperbola,  trans- 
formed to  a  pair  of  oblique  axes  whose  inclinations  to 
the  transverse  axis  are  respectively  0  and  6',  is  found 


420  ANALYTIC  GEOMETRY. 

by  changing  the  sign  of  b2  in  the  first  equation  of  p.  336, 
and  is 

(a2  sin2  0  —  b2  cos2  6)  x2  +  (a2  sin2  6r  —  b2  cos2  6'}  if 

-f-  2  (a2  sin  #  sin  6'  —  b2  cos  6  cos  0')  :n/  =  —  crb2. 

If,  then,  the  new  axes  are  the  asymptotes,  and  there- 
fore (Art.  479)  tan2  0  =  b2  :  a2  =  tan2  0',  we  shall  have 
a2  sin2  0  —  b2  cos2  6  =  0  =  a2  sin2  d'  —  b2  cos2  0'  ;  and  the 
equation  will  become 

2  (a2  sin  0  sin  6'  —  b2  cos  0  cos  6'}  xy  =  —  a2b2. 

In  this,  again,  since*  sin0  =  —  b  :  V  'a2  -f-  b'2  =  —  sin  0', 
and  cos  0  =  a  :  V  a*  +  62  =  cos  0',  it  is  evident  that  we 
have  a2  sin  6  sin  0'  —  b2  cos  0  cos  0'  =  —  2  a262  :  (a2  -f  62). 
Hence,  the  required  equation  is 


and  putting  k2  to  represent  the  constant  in  the  second 
member,  we  may  write  it  in  the  form  in  which  it  is 
usually  quoted,  namely, 

xy  =  k2. 

Corollary.  —  Hence,  the  equation  to  the  conjugate  hy- 
perbola will  be 


and,  in  the  case  of  an  equilateral  hyperbola,  we  shall 
have 

a2 


*  It  must  bo  remembered  that  9=  the  inclination  of  the  new  axis  of  x; 
and,  in  our  investigation,  the  axis  of  x  is  that  asymptote  which  corre- 
sponds to  0  =  tan"1  —  6  :  a. 


ASYMPTOTES  OF  THE  HYPERBOLA. 


421 


547.  If  <p  =  the  angle  LCR,  the  parallelogram  CMPN, 
contained  by  the  asymptotic  co- 
ordinates of  any  point  P,  will  be 

expressed  by  xy  sin  <p.  There- 
fore, from  the  equation  of  the 
preceding  article,  this  parallelo- 
gram is  equal  to  J  (a2  -\-  b2)  sin  y>. 

But  (Art.  479,  Cor.)  sin  <p  =  2ab  :  (a2  +  b2).  Hence, 
the  parallelogram  is  in  fact  equal  to  J  ab ;  and  we  have, 
as  the  geometric  interpretation  of  the  equation  xy  =  k2, 

Theorem  LVIII. — The  parallelogram  under  the  asymp- 
totic co-ordinates  of  an  hyperbola  is  constant,  and  equal  to 
half  the  rectangle  under  the  semi-axes. 

548.  Equation    to   any  Chord,   referred   to  the 
Asymptotes. — Let  x'y',  x"y"  be  the  extremities  of  the 
chord.    Then  (Art.  95),  the  equation  will  be  of  the  form 

y  — ;/      y"  —  y' 

x  —  x'  =T  x"  —  x'  ' 

Now,  since  the  extremities  of  the  chord  are  in  the  curve, 
x'  =  k2  :  y',  and  x"  =  Jc2  :  y".  Substituting  these  values, 
and  reducing,  we  get  for  the  required  equation 

y  x  -j-  x  y  =  x  y  -\-  K  (\j ; 

or,  after  dividing  through  by  Jc2  =  x'y'  =  x"y",  the  more 
symmetric  form 


_ 

x' 


=  1 


(2). 


549.   Let  Q  (&V)  and  S  (x"y"}  be  any  two  fixed  points  on 
an  hyperbola,  and  P  (a/3)  a  variable 
point.     Then   (Art.    548)    the   equa- 
tions to  PQ  and  PS  will  be 


y'ar-f  ay  = 


422  ANALYTIC  GEOMETRY. 

Making  y  =  0  in  each  of  these,  we  get  CM—  a  +  a/,  CT—  a  -f  x"  '. 
Hence,  MT  =  oc"  —  .r',  which  being  independent  of  «/?,  we  have 

Theorem  LIX.  —  The  right  lines  which  join  two  fixed  points  of  an 
hyperbola  to  any  variable  point  on  the  curve,  include  a  constant  portion 
of  the  asymptote. 

55O.  Equation  to  the  Tangent,  referred  to  the 
Asymptotes.  —  Assuming  that  the  point  x"y"  in  the  final 
equation  of  Art.  548  becomes  coincident  with  x'y',  we 
get,  for  the  equation  now  sought, 


551.  Equations  to  Diameters,  referred  to  the 
Asymptotes.  —  The  diameter  which  passes  through  a 
fixed  point  x'y'  (Art.  95,  Cor.  2),  is  represented  by 
y'x  —  x'y  =  0  ;  and  this  equation,  when  x'y'  is  on  the 
curve,  may  be  written  (Art.  546) 


The  equation  to  the  diameter  conjugate  to  x'y'  must 
have  0  for  its  absolute  term  (Art.  63)  ;  and,  as  the 
diameter  is  parallel  to  the  tangent  at  x'y'  ,  the  variable 
part  of  its  equation  (Art.  98,  Cor.)  must  be  identical 
with  that  of  the  tangent.  Hence,  the  equation  is 

J+|-0  (2). 

The  transverse  axis  bisects  the  angle  between  the 
asymptotes,  and  therefore,  at  its  extremity,  x1  --  y'. 
Hence,  the  equations  to  the  axes,  referred  to  the 
asymptotes,  are 

x  —  y  =  0,       *  +  y  =  0  (3). 


ASYMPTOTES  OF  THE  HYPERBOLA. 


423 


5*52.  Eliminating  between  (2)  of  the  preceding  article, 
and  the  equation  to  the  conjugate  hyperbola,  we  get,  for 
the  co-ordinates  of  the  extremity  of  the  diameter  conju- 
gate to  x'y', 


x  = 


553.  From  the  equation 
to  the  tangent  (Art.  550), 
we  get,  by  making  y  and  x 
successively  equal  to  zero, 
CT  =  2x',  and  OS  =  2/. 
Hence,  P  is  the  middle  point 
of  ST\  and  we  have 


Theorem  LX,  —  The  portion  of  the  tangent  included 
between  the  asymptotes,  is  bisected  at  the  point  of 
contact. 

Corollary.  —  Since  (Art.  542)  S  is  a  vertex  of  the  par- 
allelogram formed  on  the  conjugate  semi-diameters  CP 
and  CD,  we  have  PS^CD.  Hence,  8T=ZPS=Diy. 
That  is,  The  segment  cut  from  the  tangent  by  the  asymp- 
totes is  equal  to  the  diameter  conjugate  to  the  point  of 
contact. 

Remark.  —  Theorem  LX  might  have  been  obtained  geometrically, 
as  a  corollary  to  Theorem  L1V. 

554.   From  the  preceding  article,  we  at  once  obtain 


since  4  xryf  =  W.     That  is, 

Theorem  LXI.  —  The  rectangle  under  the  intercepts  cut 
off  upon  the  asymptotes  by  any  tangent  is  constant,  and 
equal  to  the  sum  of  the  squares  on  the  semi-axes. 

An.  Ge.  .39. 


424  ANALYTIC  GEOMETRY. 

555.  For  the  area  of  the  triangle  SCT,  we  have 


since  2  xfy'  =  2k2,  and  sin  (p  =  2ab  :  (a2  -j-  b2).     Hence, 

Theorem  LXII. — The  triangle  included  between  any 
tangent  and  the  asymptotes  is  constant,  and  equal  to  the 
rectangle  under  the  semi-axes. 

556.  The  equations  to  the  tangents  at  the  extremi- 
ties of  two  conjugate  diameters  (Arts.  550,  552)  are 

x         y        ~       x         11 

|         \J  O  t/  O 

x'  "h  y'  =  ?  "     y'  = 

Adding  these  together,  we  get  x  =  0,  the  equation  to 
the  line  CL.     Hence, 

Theorem  LXIII — Tangents  at  the  extremities  of  conju- 
gate diameters  meet  on  the  asymptotes. 

557.  The  equation  to  Q'Q,  an  ordinate  to  any  diam- 
eter   VD,    will    only    differ 

from  that  of  the  conjugate 
diameter  V'D'  by  some  con- 
stant, which  we  may  call  2<?. 
The  equation  will  therefore 
[Art.  551,  (2)]  be 


Now  this,  when  combined  with  the  equation  to  FT), 


gives  x  =  ex*,  y  —  cyf  as  the  co-ordinates  of  M,  the  point 
in  which  Q'Q  cuts  VD.  But  the  intercepts  of  Q'Q  upon 
the  asymptotes  are  obviously  CQf  =  2cx',  CQ  =  2cy'  '. 
Hence,  M  is  the  middle  point  of  Q'Q;  or,  we  have 


ASYMPTOTES  OF  THE  HYPERBOLA. 


425 


Theorem  LXIV. — The  segments  formed  by  the  asymptotes 
upon  an  ordinate  to  any  diameter  are  equal. 

Corollary  1. — By  the  definition  of  a  diameter,  M  is  the 
middle  point  of  P'P;  so  that  PQ  =  P'Qf,  and  we  get 
the  property:  The  portions  of  any  chord  that  are  inter- 
cepted between  the  curve  and  the  asymptotes,  are  equal. 

Corollary  2. — We  can  now  readily  solve  the  problem, 
Given  the  asymptotes  and  one  point,  to  form  the  curve. 
Let  CL,  CR  be  the  given  asymptotes, 
and  P  the  given  point.  Through  P 
draw  any  right  line  Q'Q,  cutting  the 
asymptotes  in  Q'  and  Q.  On  its  longer 
segment,  lay  off  Q 1  equal  to  the  shorter 
segment  PQr :  then  will  (1)  be  a  point 
on  the  curve,  by  Cor.  1  above.  In  the 
same  manner,  other  points,  (2),  (3), 
etc.,  may  be  obtained;  and,  when 
enough  are  found,  the  curve  can  be 
drawn  through  them.  The  given  point  P  may  be  any 
point  of  the  curve ;  but  in  practice  it  is  usually  the 
vertex,  and  is  so  represented  in  the  diagram. 

558.  The  equation  to  any  chord  Q'Q,  in  terms  of  the  extremity 
x'tf  of  its  bisecting  diameter  FZ>,  being  (Art.  557) 


the  abscissas  of  P/  and  P,  the 
points  in  which  it  cuts  the  curve, 
will  be  found  by  eliminating  be- 
tween this  equation  and  xy=  k2. 
But,  as  D  is  on  the  curve,  x'y'  = 
k2 :  whence,  by  combining  with  xy  = 


426 


ANALYTIC  GEOMETRY. 


Substituting  this  value  in  the  equation  to  the  chord,  we  get 

^7  +  —  =  2c    .'.    a:2  —  2cx'x=  —  a/2 
as  the  quadratic  determining  the  required  abscissas.     Hence, 


(Px)     x  =  x'(c+V^=:\),     (P)     x  =  x'(c—V~cT=~\). 

Now,  $  being  the  angle  between  the  asymptotes,  and  0  the  in- 
clination of  tyQ,  the  distance  from  Q  to  any  other  point  on 
is  determined  (Arts.  101,  Cor.  2;  102)  by  the  formula 

_  x  —  xl  _  (x  —  ;TI)  sin  0 


h  sin  (<j)  —  6) 

Now  the  Xi  of  Q  =  0  ;  and  (Art.  553,  Cor.  )  sin  0  :  sin  (<p  —  0)  =  b/:  x', 
where  a/  is  the  abscissa  of  Z>,  and  I/  the  semi-diameter  conjugate 
to  D.  Hence, 


P/Q  =  6/(c+ l/c2  —  1),       QP  =  6x(c  —  1/c-—  1). 
Therefore,  PXQ  .  QP  =  b/2;  and  we  have 

Theorem  LXV. —  jTAe  rectangle  under  the  segments  formed  \ipnn 
parallel  chords  by  either  asymptote  is  constant,  and  equal  to  the  square 
on  the  semi-diameter  parallel  to  the  chords. 

v.  AREA  OF  THE  HYPERBOLA. 

559.  The  area  of  the  segment  ALMP,  included 
between  the  curve,  the  asymptote,  the  ordi- 
nate  of  the  vertex,  and  the  ordinate  of  any 
given  point  P,  is  by  general  consent  called 
the  area  of  the  hyperbola.  Its  value  may 
be  determined  as  follows  :  * 

Let  x'  =  the  abscissa  CM  of  the  point  P 
to  which  the  area  is  to  be  computed.  Since 
the  co-ordinates  of  the  vertex  are  equal  to 
each  other,  we  shall  have  (from  the  equa- 
tion x  =  W  CL  —  k. 


*See  Hymers'  Conic  Sections,  p..  121,  3d  edition. 


AREA  OF  THE  HYPERBOLA.  427 

It  is  customary  to  take  the  quantity  k  as  the  unit  in 
this  computation.  Adopting  this  convention,  let  the 
distance  LMbe  so  subdivided  at  n  points  R,  S9  .  .  .  ,  M, 
that  the  abscissas  CL,  CR,  CS,  .  .  .  ,  CM  may  increase 
by  geometric  progression.  Then,  if  OR  =  x,  we  shall  have 


Thus  x'  =  xn  ;  or,  x  =  xf»  :  so  that,  as  n  increases,  x 
diminishes,  and  converges  toward  1  as  n  converges  to 
infinity.  Now,  at  R,  $,  .  .  .  ,  If,  erect  n  ordinates,  and 
form  n  corresponding  parallelograms  RA,  Sa,  .  .  .  ,  Mo, 
situated  as  in  the  figure.  Then 

area  RA  —  RL  .  LA  sin  0  =  (  C72  —  CL)  LA  sin  0  =  (x  —  1  )  sin  c&, 
"       Sa  =  SR  .  Ra  sin  <j>  =  (x2  —  x)  —  sin  0  =  (x  —  1)  sin  $, 

u       Te=TS.Se    sin  0=  (x3—  x1)  ^  sin  ^  =  (.r  —  1)  sin^, 

and  so  on  for  the  whole  series  of  n  parallelograms. 
Hence,  replacing  x  by  its  value  a/"",  and  putting  2'  = 
the  sum  of  the  n  parallelograms,  we  get 

I  =  n  (x''*  —  1)  sin  (p. 

But  an  inspection  of  the  diagram  shows  that  the 
greater  the  number  of  the  parallelograms,  the  more 
nearly  does  the  sum  of  their  areas  approach  the  area 
ALMP.  And  since  x,  the  ratio  of  the  successive 
abscissas,  tends  to  1  as  n  tends  to  op,  we  can  make  the 
number  of  the  equal  parallelograms  as  great  as  we 
please  ;  in  other  words,  the  true  value  of  the  area 
ALMP  is  the  limit  toward  which  2'  converges  as  n 
converges  to  oo.  Hence, 

area  ALMP  =  n  (a/n—  1)  sin  <£  =  n  [{  1  +  (x7—  1)  }n—  1]  sin  <? 


428  ANALYTIC  GEOMETRY. 

area  ALMP  =  {  (xf—  1)  +  ~(^  - 1)  (x/—  I)2 


.r'-l       (.r'-l)'       (^-l)3      (a;7-  1)4  . 

~~  ~"  ~~~  ~~  '  '  '     S1 


l 
r    I 

Now  [Alg.,  373,  (5)]  the  series  in  the  braces  denotes  the 
Naperian  logarithm  of  x'.  Therefore,  calling  this  loga- 
rithm lx',  and  the  hyperbolic  area  A,  we  obtain 

A  =  sin  (f>  .  I  x'. 

But  (Alg.,  376)  sin  <p  .  I  x'  =  the  logarithm  of  xr  in  a 
system  whose  modulus  =  sin  (p.  Hence, 

Theorem  LXVI.  —  The  area  of  any  hyperbolic  segment 
is  equal  to  the  logarithm  of  the  abscissa  of  its  extreme  point, 
taken  in  a  system  whose  modulus  is  equal  to  the  sine  of  the 
angle  betiveen  the  asymptotes. 

Corollary,  —  In  an  equilateral  hyperbola,  since  <p  —  90°, 
sin  <p  =  1  ;  and  we  get  A  =  lxr  .  That  is,  The  area  of  an 
equilateral  hyperbola  is  equal  to  the  Naperian  logarithm 
of  the  abscissa  of  the  extreme  point. 

For  this  reason,  Naperian  logarithms  are  called  hyper- 
bolic. But,  as  we  have  just  seen,  the  title  belongs  with 
equal  propriety  to  logarithms  with  any  modulus. 

EXAMPLES  ON  THE  HYPERBOLA. 

1.  Prove  that  the  middle  points  of  a  series  of  parallels  inter- 
cepted between  an  hyperbola  and  its  conjugate,  lie  on  the  curve 


_ 

a"       I'* 

2  Find  the  several  loci  of  the  centers  of  the  circles  inscribed 
and  escribed  to  the  triangle  F'PF,  F/  and  F  being  the  foci  of  any 
hyperbola,  and  P  any  point  on  the  curve. 


EXAMPLES  ON  THE  HYPERBOLA.  429 

3.  An  ellipse  and  a  pair  of  conjugate  hyperbolas  are  described 
upon  the  same  axes,  and,  at  the  points  where  any  line  through  the 
center  meets  the  ellipse  and  one  of  the  hyperbolas,  tangents  are 
drawn :  find  the  locus  of  their  intersection. 

4.  In  any  triangle  inscribed  in  an  equilateral  hyperbola,  the 
three  perpendiculars  from  the  vertices  to  the  sides,  converge  in  a 
point  upon  the  curve. 

5.  To  determine  the  hyperbola  which  has  two  given  lines  for 
asymptotes,  and  passes  through  a  given  point. 

6.  Between  the  sides  of  a  given  angle,  a  right  line  moves  so  as 
to  inclose  a  triangle  of  constant  area :   the  locus  of  the  center  of 
gravity  in  the  triangle  is  the  hyperbola  represented  by 

9  xy  sin  $  =  2&2, 

where  <b  =  the  given  angle,  and  k2  =  the  constant  area. 

7.  QQ/  is  a  double  ordinate  to  the  axis  major  A/A  of  an  ellipse ; 
QA,  A' Qf  are  produced  to  meet  in  P:  find  the  locus  of  P. 

8.  To  a  series  of  con  focal  ellipses,  tangents  are  drawn  having  a 
constant  inclination  to  the  axes :  the  locus  of  the  points  of  contact 
is  an  hyperbola  concentric  with  the  ellipses. 

9.  The  radius  of  the  circle  which  touches  an  hyperbola  and  its 
asymptotes,  is  equal  to   that  part  of  the  latus   rectum  produced 
which  is  intercepted  between  the  asymptote  and  the  curve. 

10.  About  the  focus  of  an  hyperbola,  a  circle  is  described  with 
a  radius  equal  to  the  conjugate  semi-axis,  and  tangents  are  drawn 
to  it  from  any  point  on  the  curve :  their  chord  of  contact  is  tangent 
to  the  inscribed  circle. 

11.  Tangents  to  an  hyperbola  are  drawn  from  any  point  on  either 
branch  of  the  conjugate  curve:  their  chord  of  contact  touches  the 
opposite  branch. 

12.  In  any  equilateral  hyperbola,  let  <i>  =  the  inclination  of  a 
diameter  passing  through  any  point  P,  and  tf  =  that  of  the  polar 
of  P,  the  transverse  axis  being  the  axis  of '  oc:  then  will 

tan  $  tan  <j>'  =  1. 

13.  The  circle  which  passes  through  the  center  of  an  equilateral 
hyperbola  and  any  two  points  A  and  B,  passes  also  through  the 
intersection  of  two  lines  drawn  the  one  through  A  parallel  to  the 
polar  of  J5,  and  the  other  through  B  parallel  to  the  polar  of  A. 


430  ANALYTIC  GEOMETRY. 

14.  The  locus  of  a  point  such  that  the  rectangle  under  the  focal 
perpendiculars  upon  its  polar  with  respect  to  a  given  ellipse  shall 
be  constant,  is  an  ellipse  or  an  hyperbola  according  as  the  foci  are 
on  the  same  side  or  on  opposite  sides  of  the  polar. 

15.  A  line  is  drawn  at  right  angles  to  the  transverse  axis  of  ai5. 
hyperbola,  meeting  the  curve  and  its  conjugate  in  P  and  Q:  show 
that  the  normals  at  P  and  Q  intersect  upon  the  transverse  axis. 
Also,  that  the  tangents  at  P  and  Q  intersect  on  the  curve 


16.  Given  two  unequal  circles:    the  locus  of  the  center  of  the 
circle  which  touches  them  both  externally,  is  an  hyperbola  whose 
foci  are  the  centers  of  the  given  circles. 

17.  Every  chord  of  an  hyperbola  bisects  the  portion  of  either 
asymptote  included  between  the  tangents  at  its  extremities. 

18.  If  a  pair  of  conjugate  diameters  of  an  ellipse  be  the  asymp- 
totes of  an  hyperbola,  to  prove  that  the  points  of  the  hyperbola  at 
which  its  tangents  will  also  touch  the  ellipse,  lie  on  an  ellipse  con- 
centric and  of  the  same  eccentricity  with  the  given  one. 

19.  A  tangent  is  drawn  at  a  point  P  of  an  hyperbola,  cutting 
the  asymptote  CY  in  E;   from  E  is  drawn  any  right  line  EKH 
cutting  one  branch  of  the  curve  in  A"  //;   and  Kk,  PM,  Hh  are 
drawn  parallel  to   CY  cutting  the  asymptote   CK  in  k,  M,  h:   to 
prove  that 


20.  Three  hyperbolas  have  parallel  asymptotes:  show  that  the 
three  right  lines  which  join  two  and  two  the  intersections  of  the 
hyperbolas,  meet  in  one  point. 


CHAPTER    FIFTH. 

THE    PARABOLA, 
i.  THE  CURVE  REFERRED  TO  ITS  Axis  AND  VERTEX. 

56O.   In   discussing    the   Parabola,   we    shall   find    it- 
most  convenient  to  transform  its  equation,  as  found  in 


PROPERTIES  OF  THE  PARABOLA.  431 

Art.  181,  to  a  new  set  of  reference-axes.  Before  doing 
so,  however,  we  may  deduce  an  important  property, 
which  will  enable  us  to  give  the  constant  p,  involved  in 
that  equation,  a  more  significant  interpretation. 


THE    AXIS. 


561.  Making  y  =  0   in  the   equation 
mentioned,  namely,  in 


we  obtain  x  =  OA=p.  Now  (Art.  181) 
2p  =  OF:  hence,  OA  =  J  OF',  or,  we 
have 

Theorem  I. — In  any  parabola,  the  vertex  of  the  curve 
bisects  the  distance  between  the  focus  and  the  directrix. 

Corollary. — Whenever,  therefore,  the  constant  p  pre- 
sents itself  in  a  parabolic  formula,  we  may  interpret  it 
as  denoting  the  distance  from  the  focus  to  the  vertex  of 
the  curve. 

Remark. — We  might  also  infer  the  theorem  of  this  article  directly 
from  the  definition  of  the  curve  in  Art.  179. 

562.  Since  AF  is  thus  equal  to  p,  the  distance  of 
the  focus  from  the  vertex  wrill  converge  to  0  whenever 
%p  =  OF  converges  to  that  limit,  but  will  remain  finite 
as  long  as  2p  remains  so.  In  other  words  (since  we  may 
take  the  focus  F  as  close  to  the  directrix  D'D  as  we 
please),  the  focus  may  approach  infinitely  near  to  the 
vertex,  but  can  not  pass  beyond  it.  Hence, 

Theorem  II. —  The  focus  of  a  parabola  falls  within  the 

curve. 


563.  Let  us  now  transform  the  equation  u2  =  4p  (x—p), 
by  moving  the  axis  of  y  parallel  to  itself  along  OF  to 


432  ANALYTIC  GEOMETRY. 

the  vertex  A.     This  we  accomplish  (Art.  55)  by  putting 
x  -f  p  for  x,  and  thus  get 

f  =  4px. 

564.  This  equation  asserts  that  the  ordinate  of  the 
Parabola  is  a  geometric  mean  of  the  abscissa  and  four 
times  the  focal  distance  of  the  vertex. 
It  therefore  leads  directly  to  the  fol- 
lowing construction  of  the  curve  by  ^;:'-l. 
points,  when  the  focus  and  the  vertex  L  ---  -< 
are  given  :  —  Through  the  focus  F  draw 
the  axis  M'A,  and  produce  it  until 
AB  =  4AF.  Through  the  vertex  A 
draw  ADr  perpendicular  to  the  axis.  At  any  convenient 
points  on  the  axis,  erect  perpendiculars  of  indefinite 
lengths,  as  MP,  M'P*;  and  upon  the  distances  BM, 
BM',  etc.,  as  diameters,  describe  circles  BDM,  BD'M', 
etc.  From  D,  D',  .  .  .  ,  where  these  circles  cut  the  per- 
pendicular AD  ',  draw  parallels  to  the  axis  :  the  points 
P.  P',  .  .  .  ,  in  which  these  meet  the  perpendiculars  MP, 
M'P1,  .  .  .  ,  will  be  points  of  the  parabola  required.  For 


we  shall  have  PM=  AD  =      AB.AM  = 

and  a  similar  relation  for  P'M'  and  all  other  ordinates 
formed  in  the  same  way. 


565.  The  property  asserted  by  the  equation  y*  = 
and  involved  in  the  foregoing  construction,  may  be  other- 
wise stated  as 

Theorem  III.  —  The  square  on  any  ordinate  of  a  para- 
bola is  equal  to  four  times  the  rectangle  under  the  cor- 
responding  abscissa  and  the  focal  distance  of  the  vertex. 

Corollary.  —  Since  p  is  constant  for  any  given  parabola, 
yz  will  increase  or  diminish  directly  as  x  does.  That  is, 


PARABOLA  LIMIT  OF  ELLIPSE.  433 

The  squares  on  the  ordinates  of  any  parabola  vary  as  the 
corresponding  abscissas: 

«>66.  The  double  ordinate  L'L  passing 
through  the  focus,  in  the  Parabola  also,  is 
called  the  latus  rectum.  Making  x  =  p  in 
the  equation  to  the  curve,  we  obtain  the 
value  of  FL)  namely,  y  =  2p.  Hence, 

latus  rectum  =  4p. 

Corollary. — This  result  would  lead  us  to  state  Theorem 
III  as  follows:  The  square  on  any  ordinate  is  equal  to 
the  rectangle  under  the  corresponding  abscissa  and  the 
latus  rectum. 

567.  It  is  now  important  to  show  that  a  certain 
relation  in  form  exists  between  the  Parabola  and  the 
Ellipse,  such  that  we  may  consider  a  parabola  as  the 
limiting  shape  to  which  an  ellipse  approaches  when  we 
conceive  its  axis  major  to  increase  continually,  while  its 
focus  and  the  adjacent  vertex  remain  fixed.  By  estab- 
lishing this,  we  shall  be  enabled  to  bring  the  symbol  e, 
arbitrarily  written  =  1  in  Art.  184,  under  the  conception 
which  gives  it  meaning  in  the  other  two  conies. 

The  equation  we  are  now  using  for  the  Parabola  being 
referred  to  the  vertex,  we  must  refer 
the  Ellipse  to  its  vertex,  if  we  desire 


to  exhibit  the  relation  mentioned. 
Transforming,  then,  the  equation  of 
Art.  147  by  putting  x  —  a  for  x,  we 
get,  as  the  equation  to  the  Ellipse  referred  to  its  vertex  F, 


434  ANALYTIC  GEOMETRY. 

Putting  p  as  an  arbitrary  symbol  for  the  distance  VF 
between  the  vertex  and  the  adjacent  focus,  we  have 
(Art.  151)  p  =  a  —  l/V  —  62 :  whence 


and  the  above  equation  becomes 


Suppose,  now,  that  the  distance  VF  remains  fixed, 
while  the  whole  axis  major  VA  increases  to  infinity  : 
in  the  limit,  where  a  =  GO,  we  get 

if  =  4px, 

the  equation  to  the  Parabola  :  which  proves  our  propo- 
sition. 

Corollary  1.  —  Since  we  may  thus  regard  a  parabola  as 
an  ellipse  with  an  infinitely  long  axis,  that  is,  with  a 
center  infinitely  distant  from  its  vertex  and  its  focus,  it 
deserves  to  be  considered  whether  the  Parabola  has  any 
element  analogous  to  the  so-called  circumscribed  circle 
of  the  Ellipse.  To  settle  this  point,  we  only  need  to 
consider,  that,  if  a  tangent  be  drawn  to  a  circle,  and  the 
radius  of  the  circle  be  then  continually  increased  without 
changing  the  point  of  contact,  the  circle  will  tend  more 
and  more  nearly  to  coincidence  with  its  tangent  the 
greater  the  radius  becomes;  so  that,  if  we  were  to 
suppose  the  radius  infinitely  great,  the  circle  would 
become  straight  by  actually  coinciding  with  the  tangent. 
If,  then,  we  draw  at  the  vertex  of  an  ellipse  a  common 


ECCENTRICITY.  435 

tangent  to  the  curve  and  its  circumscribed  circle,  and 
subject  the  axis  major  to  continual  increase  under  the 
conditions  which  will  cause  the  curve  to  assume  the  form 
of  a  parabola  in  the  limit  where  a  =  GO,  the  circumscribed 
circle  will  continually  approach  the  common  tangent  as 
its  center  recedes  from  the  vertex,  and,  in  the  limit  where 
the  ellipse  vanishes  into  a  parabola,  will  coincide  with  the 
tangent.  We  learn,  then,  that  the  Parabola  has  an  ana- 
logue of  the  circumscribed  circle;  that  this  analogue  is 
in  fact  the  tangent  of  the  curve  at  its  vertex ;  and  that 
it  is  also  the  line  used  as  the  axis  of  y  in  the  equation 
?/2  —  4^  since  this  line  and  the  tangent  are  both  per- 
pendicular to  the  axis  of  the  curve  at  its  vertex,  and 
must  therefore  coincide.  All  these  results  will  soon  be 
confirmed  by  analysis. 

Corollary  2, — But,  as  a  point  of  greater  importance, 
the  relation  established  above  enables  us  to  assert  that 
the  symbol  e  =  1  denotes  the  eccentricity  of  the  corre- 
sponding parabola.  For,  in  the  Ellipse,  we  have 

a2  — 62_          b2  . 

e -1  ~      > 


and  if  in  this  we  make  a  =  GO,  or  suppose  the  curve  to 
become  a  parabola,  we  get  e  =  l.  That  is,  we  may  con- 
sider a  parabola  to  be  an  ellipse  in  which  the  eccentricity 
has  reached  the  limit  1.  Moreover,  in  the  view  taken 
of  this  subject  in  the  corollaries  to  Arts.  359,  456,  the 
condition  e  =  1  corresponds  to  that  ellipse  which  has  so 
far  deviated  from  the  curvature  of  its  circumscribed  circle 
as  to  vanish  into  the  right  line  that  forms  its  axis  major. 
Now,  when  we  recollect  (Art.  195,  Cor.)  that  a  right  line 
is  a  particular  case  of  the  Parabola,  and  that,  in  reaching 
the  limit  1,  e  has  assumed  a  value  fixed  and  the  same 


436  ANALYTIC   GEOMETRY. 

for  all  parabolas,  it  becomes  evident  that  we  may  con- 
sider the  condition  e  =  1  as  marking  that  stage  of  devi- 
ation from  circularity  which  characterizes  the  Parabola, 
and  therefore,  in  connection  with  this  curve  also,  appro- 
priately call  e  the  eccentricity. 

Hence,  the  name  parabola  (derived  from  the  Greek 
xapafidttsw,  to  place  side  by  side,  to  make  equal]  may  be 
taken  as  signifying,  that,  in  the  curve  which  it  denotes, 
the  eccentricity  is  equal  to  unity. 

In  closing>  this  article,  we  would  again  direct  the 
student's  attention  to  the  fact,  that  all  parabolas  have 
the  same  eccentricity. 


Let  p  =  FP,  the  focal  distance  of  any  point  on 
a  parabola.  Then,  by  the  definition 
of  the  curve  (BD  being  the  directrix), 
FP  =  PD.  Also,  from  the  diagram, 
PD  =  BM=  BA  -f  AM.  Now,  AM 
=  x,  and  (Art.  561)  BA  =p.  Hence, 

p  =p  +  x. 
That  is, 

Theorem  IV.  —  The  focal  radius  of  any  point  on  a  par- 
abola is  a  linear  function  of  the  corresponding  abscissa. 

Remark.  —  This  expression  for  p  is  similar  to  those 
found  in  the  case  of  the  Ellipse  and  of  the  Hyperbola 
(Arts.  360,  457),  and  is  called  the  Linear  Equation  to 
the  Parabola. 

5159.  The  methods  of  drawing  the  curve,  given  in 
Arts.  179,  564,  have  already  familiarized  the  reader 
with  the  figure  of  the  Parabola,  and  suggested  a  tol- 
erably clear  conception  of  its  details.  Let  us  now  see 
how  the  equation 


FIGURE  OF  THE  PARABOLA.  437 

verifies    and    completes    the    impressions,  made    by   the 
diagrams  : 

I.  No  point  of  the  curve  lies  on  the  left  of  the  perpendicular 
to  the  axis,  drawn  through  the  vertex.  For  this  line  is  the  axis 
of  y;  and,  if  we  make  x  negative  in  the  equation,  the  resulting 
values  of  y  are  imaginary. 

II.  But  the  curve  extends  to  infinity  on  the  right  of  the  per- 
pendicular mentioned,  both  above  and  below  the  axis.  For  y  is 
real  for  every  possible  positive  value  of  x. 

III.  The  curve  is  symmetric  to  the  axis.  For  there  are  two 
values  of  y,  numerically  equal  but  opposite  in  sign,  corresponding 
to  every  value  of  x. 

57O.  The  Parabola,  then,  differs  from  the  Ellipse, 
and  resembles  the  Hyperbola,  in  having  infinite  con- 
tinuity of  extent.  There  is,  however,  a  marked  dis- 
tinction between  its  infinite  branch  and  the  infinite 
branches  of  the  Hyperbola,  which  calls  for  a  more 
minute  examination. 

In  the  first  place,  the  limiting  forms  to  which  the  two 
curves  tend  are  essentially  different :  that  of  the  Hyper- 
bola (Art.  176)  being  two  intersecting  right  lines,  which 
pass  through  its  center ;  while  that  of  the  Parabola 
(Arts.  192;  195,  Cor.)  is  two  parallel  right  lines,  or,  in 
the  extreme  case,  a  single  right  line.  This,  of  itself, 
indicates  a  difference  in  the  nature  of  the  curvature  in 
these  two  conies. 

Secondly,  the  branches  of  the  Hyperbola,  in  receding 
from  the  origin,  tend  to  meet  the  two  lines  called  the 
asymptotes  in  two  coincident  points  at  infinity  (Arts. 
539,  543).  Now,  if  we  seek  the  intersections  of  any 
right  line  with  a  parabola,  by  eliminating  between  the 
equations  y  =  mx  -f  b  and  y2  =  4px,  we  find  that  their 
abscissas  are 


(2p  —  ml)  ±V/p- 
x  = 


438  ANALYTIC  GEOMETRY. 

In  order,  then,  that  these  intersections  may  be  coinci- 
dent, we  must  have  inb  =  p :  in  which  case,  for  the 
abscissas,  we  get 

?.*£  (1)' 

and,  for  the  equation  to  the  given  line, 

y  —  mx  -f-  —  ;    or,  m2x  —  my  -\-p  =  Q        (2) . 


If,  now,  the  coincidence  takes  place  at  infinity,  we  shall 
have,  from  (1),  jn  =  Q-,  and  (2)  will  assume  the  form 
(Art.  110) 

C=Q. 

That  is,  any  right  line  that  tends  to  meet  a  parabola  in 
two  coincident  points  at  infinity,  is  situated  altogether  at 
infinity ;  or,  what  is  the  same  thing,  no  parabola  has  any 
tendency  to  approach  a  finite  right  line  in  the  manner 
characteristic  of  the  Hyperbola. 

DIAMETERS. 

571.  Equation  to  any  Diameter. — In  a  system  of 
parallel  chords  in  a  parabola,  let  6  be  the  common  in- 
clination to  the  axis,  xy  the  middle  point  of  any  member 
of  the  system,  and  x'y'  the  point  in  which  the  chord  cuts 
the  curve.  We  have  (Art.  101,  Cor.  3) 

x'  =  x  —  Z  cos  #,      yt=y  —  Zsin#. 
Hence,  as  x'yf  is  on  the  parabola  mentioned, 
(y  —  I  sin  d)2  =  4p(x  —  l  cos  6). 
That  is,  for  determining  Z,  we  get  the  quadratic 
Z2sin2  0  —  2  (?/  sin  0  —  2p  cos  6)  l  =  4px  —  y\ 


DIAMETERS  OF  THE  PARABOLA.  439 

But  as  xij  is  the  middle  point  of  a  chord,  the  co-efficient 
of  I  vanishes  (Alg.,  234,  Prop.  3d),  and  the  locus  of  the 
middle  point  is  therefore  represented  by  the  equation 

y  sin  0  —  2p  cos  0  =  0. 

Hence,  the  required  equation  to  any  diameter  is 
y  =  2p  cot  0. 

572.  Since  p  is  fixed  for  any  given  parabola,  and  0 
for  any  given  system  of  parallel  chords,  this  equation 
is  of  the  form 

y  =  constant. 

Now  such  an  equation  (Art.  25)  denotes  a  parallel  to  the 
axis  of  x.  Hence, 

Theorem  V. — Every  diameter  of  a  parabola  is  a  right 
line  parallel  to  its  axis. 

Corollary  1. — From  this,  we  at  once  infer:  All  the 
diameters  of  a  parabola  are  parallel  to  each  other. 

Corollary  2. — The  constant  value  of  y 
in  the  above  equation,  being  dependent 
on  the  arbitrary  angle  0,  is  itself  arbi- 
trary. The  converse  of  our  theorem  is 
therefore  true,  and  we  have :  Every  right 
line  drawn  parallel  to  the  axis  of  a  par- 
abola is  a  diameter. 

Remark. — The  directrix  being  perpendicular  to  the  axis,  we 
might  define  a  diameter  of  a  parabola  as  any  right  line  drawn 
perpendicular  to  the  directrix.  The  diagram  illustrates  diameters 
from  either  point  of  view. 

573.  If  we  write  (as  we  may)  the  equation  y  =  constant 
in  the  form 

y  =  Ox  -f  6, 
An.  Ge.  40. 


440  ANALYTIC  GEOMETRY. 

and  then  eliminate  y  between  it  and  y1  =  tyx,  we  get, 
for  determining  the  intersections  of  a  parabola  with  its 
diameter,  the  relation  Ox2  -j-  Ox  -f  b2  =  4px  ;  or, 


Ox2  —  4px  -\-b2=Q. 
Now  (Alg.,  238)  the  roots  of  this  quadratic  are 


x  —  GO  : 


whence,  observing  the  form  of  these  roots,  we  have 

Theorem  VI.  —  Every  diameter  of  a  parabola  meets  the 
curve  in  two  points,  one  finite,  the  other  at  infinity. 

Eemark.  —  The  meaning  of  this  theorem,  in  ordinary  geometric 
Ian  future,  is,  of  course,  that  a  diameter  only  meets  the  curve  in  the 
one  finite  point  whose  abscissa  =  62  :  4p.  The  argument  and  the 
phraseology  adopted  here  are  only  used  for  the  purpose  of  com- 
pleting analogies,  and  will  seem  less  forced  when  we  approach  the 
subject  from  a  more  generic  point  of  view. 

574.  From  the  last  theorem,  it  follows  that  no  chord 
of  a  parabola  can  be  parallel  to  the  axis,  and,  therefore, 
that  no  diameter  can  bisect  a  system  of  chords  parallel 
to  a  second  diameter.     We  thus  learn,  that,  in  the  Par- 
abola, the  conception  of  conjugate  diameters  vanishes  in 
the  parallelism  of  all  diameters. 

THE    TANGENT. 

575.  Equation  to  any  Chord  —  If  x'y',  x"y"  be  the 

extremities  of  any  chord  in  a  parabola,  we  shall  have 
y'*  =  ±px'  and  y"*  =  4px".  Hence,  y"2—y'2  =  4p  (x"—x'\ 
and  we  get 

y"-y'        *P 

x"-*  -  y'  +  y"' 


TANGENT  OF  THE  PARABOLA.  441 

Substituting  this  value  for  the  second  member  of,  the 
equation  in  Art.  95,  we  obtain  the  equation  now  re- 
quired, namely, 

y  —  y'  .    _4p_ 

x  —  x'      .:   i/f  -f-  y"  ' 

576.  Equation  to  the  Tangent.  —  Making  y"  =  y' 
in  the  preceding  equation,  reducing,  and  recollecting 
that  y'2  —  4pxf,  we  get 


577.  Condition  that  a  Right  Line  shall  touch  a 
Parabola.  —  We  have  seen,  in  the  second  part  of  Art. 
570,  that  y  —  mx  -f-  b  will  meet  the  parabola  y2  —  4px  in 
two  coincident  points,  whenever  mb—p.    The  condition 
now  required  is  therefore 

*=£. 

m 

Corollary,  —  Every  right  line,  then,  whose  equation  is 
of  the  form 

y  =  mx  -f-  — 
m 

is  a  tangent  to  the  parabola  y2=4px.  We  have  here 
another  instance  of  the  so-called  Magical  Equation  to  the 
Tangent. 

578.  Problem,  —  If  a   tangent  to  a  parabola  passes 
through  a  fixed  point,  to  find  the  co-ordinates  of  contact. 

Putting  x'y'  for  the  required  point  of  contact,  and 
x"y"  for  the  fixed  point  through  which  the  tangent 
passes,  we  have  (Arts.  563,  576) 


442  ANALYTIC  GEOMETRY. 

Solving  these  conditions  for  x'  and  y',  we  get 

*'=& 


y'  =  y"  ±Vy"'2  - 

Corollary.  —  These  values  indicate  that  from  any  given 
point  two  tangents  can  be  drawn  to  a  parabola  :  real 
when  y"2  —  4pxff  >  0,  that  is,  when  the  point  is  ivitliout 
the  curve;  coincident  when  y"2  —  4jwc"  =  0,  that  is,  when 
the  point  is  on  the  curve  ;  imaginary  when  y"2  —  4pz"<  0, 
that  is,  when  the  point  is  within  the  curve. 

579o  Definition.  —  The  halves  of  the  chords  which  any 
diameter  of  a  parabola  bisects,  are  called  the  ordinates 
of  the  diameter.  The  term  is  also  applied  at  times  to 
the  entire  chords,  or  to  the  right  lines  formed  by  pro- 
longing them  indefinitely. 

«58Oo  Let  x'yf  be  the  extremity  of  any  parabolic  di- 
ameter. Then,  from  Art.  571,  ?/'—  2pcot#;  and  we  get, 
for  determining  the  angle  6  which  the  ordinates  of  the 
diameter  make  with  the  axis, 


y' 

Now  the  equation  to  the  tangent  at  x'y'  (Art.  576)  may 
be  written 


Hence,  by  the  principle  of  Art.  78,  Cor.  1, 

Theorem  VII.  —  The  tangent  at  the  extremity  of  any 
diameter  of  a  parabola  is  parallel  to  the  corresponding 
ordinates. 


DIRECTION  OF  THE  TANGENT. 


443 


Corollary. — Hence,  further,  the  tangent  at  the  vertex 
of  the  curve  is  perpendicular  to  the  axis ;  and  we  confirm 
by  analysis  the  result  of  Art.  567,  Cor.  1,  namely,  The 
vertical  tangent  and  the  axis  of  y  are  identical. 

581.  The  equations  to  any  tangent  of  a  parabola  and 
its  focal  radius  of  contact  (Arts.  576,  95)  may  be  written 


y'y  =  2p  (x  +  x')         (PT), 
y'x+(p-x')y=py<      (FP). 

For  the  angle  FPT  between  these 
lines,  we  therefore  have  (Art.  96, 
Cor.  1) 


y' 


But,  by  Art.  580,  this  is  also  the   value  of  tan  QPT. 
Hence  FPT  =  QPT',  and  we  get 

Theorem  VIII. —  The  tangent  of  a  parabola  bisects  the 
internal  angle  between  the  diameter  and  focal  radius  drawn 
to  the  point  of  contact. 

Corollary  1. — To  draw,  then,  a  tangent  to  a  parabola 
at  any  point  P,  we  form  the  focal  radius  PF,  and  the 
diameter  RP.  We  next  prolong  RP  until  PQ  equals 
PF,  join  QF,  and  draw  PT  perpendicular  to  QF:  it 
will  be  the  tangent  required,  by  virtue  of  the  theorem 
just  proved,  and  the  isosceles  triangle  FPQ. 

Corollary  2,— Since  SPR=QPT=FPT,  all  rays  that 
strike  the  concave  of  the  curve  in  lines  parallel  to  the 
axis  will  be  reflected  to  F,  which  is  therefore  called  the 
focus,  as  in  the  Ellipse  and  the  Hyperbola. 


444  ANALYTIC   GEOMETRY. 

582.  Making  y  =  0  in  the  equation  y'y  =  2 
we  obtain,  as  the  value  of  the  intercept 
formed  by  the  tangent  upon  the  axis  of 
a  parabola, 


The  length  of  the  intercept  is  therefore 

equal  to  that  of  the  abscissa  of  contact, 

the    sign    minus  denoting  that   it   is   measured    to    the 

left  of  the  vertex.     If,  then,  we  add  AF  =  p  to   this 

length,  we  get 


But  (Art.  568)  p  +  x'  =  FP.     Hence,  FT=FP;  and 
we  have 

Theorem  IX. — In  any  parabola,  the  foot  of  the  tangent 
and  the  point  of  contact  are  equally  distant  from  the 
focus. 

Corollary. — This  property  obviously  leads  to  the  fol- 
lowing constructions  : 

I.  To  draw  a  tangent  at  any  point  P  of  a  parabola. 
Join  the  given  point  P  with  the  focus  JF,  and  from  the 
latter  as  a  center,  with  a  radius  equal  to  PF,  describe 
an  arc  cutting  the  axis  in  T:  the  required  tangent  may 
then  be  formed  by  joining  PT. 

II.  To  draw  a  tangent  to  a  parabola  from  any  point  T 
on  the  axis.     From   the  focus  F  as  a  center,  with  the 
radius   FT,  describe   an   arc,   and  note  the  point  P  in 
which   this    cuts    the    curve:    P   will   be    the    point    of 
contact,  and  the  corresponding  tangent  may  be  formed 
by  joining  TP. 


SUBTANGENT  OF  THE  PARABOLA.  445 

583.  The  Sufrtaiigent. — For  the  length  of  the  sub- 
tangent  of  the  curve  in  the  Parabola, 

or  of  that  portion  of  the  axis  which  is 
included  between  the  foot  of  the  tan- 
gent and  that  of  the  ordinate  of  con- 
tact, we  have  TM=  TA-\-  AM;  or, 

subtan  =  x'  -f  x'  =  2xf. 

584.  Thus  TA  =  AM=  J  TM,  and  we  get  the  im- 
portant property, 

Theorem  X. —  The  subtangent  of  a  parabola  is  bisected 
in  the  vertex. 

Corollary  1. — We  can  now  construct  the  tangent  at 
any  point  of  the  curve  or  from  any  point  on  the  axis,  as 
follows :  — When  the  point  of  contact  P  is  given,  draw 
the  ordinate  PM,  and  on  the  prolonged  axis  lay  off 
AT=AM:  then,  by  the  theorem  just  proved,  T  will 
be  the  foot  of  the  tangent  at  P,  which  is  found  by 
joining  PT. 

When  the  foot  T  of  the  tangent  is  given,  lay  off  upon 
the  axis  AM --  AT,  and  erect  the  ordinate  MP.  The 
point  P  in  which  this  meets  the  curve,  will  be  the  point 
of  contact  sought;  and  the  tangent  is  obtained  by  join- 
ing TP. 

Corollary  2. — As  the  value  of  the  subtangent  is  depend- 
ent upon  the  peculiar  form  of  the  equation  to  the  Para- 
bola, the  present  theorem  is  peculiar  to  this  curve,  and 
hence  leads  to  the  following  mode  of  constructing  it, 
which  is  often  used  by  mechanics  and  draughtsmen. 

Lay  down  two  equal  right  lines  AB,  AC  making  any  convenient 
angle  with  each  other.  Bisect  them  in  E  and  F,  join  EF,  BC,  and 
draw  AX  perpendicular  to  the  latter:  Ic  will  bisect  EF  in  V,  and 
BC  in  X,  by  the  well-known  properties  of  the  isosceles  triangle. 


446 


ANALYTIC   GEOMETRY. 


Now  divide  AE  and  its  equal  EB  into  the  same  number  of  equal 

parts,  and  the   equals  AF,  FC  in  the 

same  manner:  the  whole  lines  A  5,  AC 

will  thus  be  subdivided  into  equal  parts 

at  the  points  1,  2,  3,  E,  4,  5,  6  and  6,  5, 

4,  F,  3,  2,  1.     Having  numbered  these 

points  in  reverse  order  upon  the  two    A 

lines,  as  in  the  diagram,  join  those  which 

have  the  same  numeral  :  the  resulting 

lines   will  envelope   a  parabola,  which 

we  can  approximate  as  closely  as  we 

please,  by  continually  diminishing  the 

distances  Al,  1  ...  2,  etc.    For,  by  the  construction,  V  is  the  middle 

point  of  AX;  and  the  curve  touches  all  the  lines  AE,  AC,  1  ...  1, 

6.  .  .  6,  etc.  :  hence,  with  respect  to  the  lines  AB,  AC,  which  may 

be  regarded  as  limiting  cases  of  all  the  others,  it  is  a  curve  that 

bisects  its  subtangent  in  the  vertex  ;  that  is,  a  parabola. 

From  another  point  of  view,  the  curve  here  formed  is  the  envel- 
ope of  a  line  EF,  which  moves  within  the  fixed  lines  AB,  AC  in 
such  a  manner  that  the  sum  of  the  remaining  sides  of  the  triangle 
EAF  is  constant,  being  equal  to  AB.  It  is  therefore  a  parabola, 
by  the  result  of  the  Example  solved  in  Art.  251. 

585.  Perpendicular  from  the  Foeiis  to  any  Tan- 
gent. —  For  the  length  of  the  perpendicular  from  the  focus 
(p,  0)  upon  the  line  y'y^Zp  (x  -f-  #'),  we  have  (Art.  105, 
Cor.  2) 

xf)  2p  (p  + 


= 


But  (Art.  568)  p-\-xf=p,  the  focal  distance  of  the  point 
of  contact.     Hence, 


5S6.  In  this  expression,  p  being  constant,  it  is  evident 
that  P2  will  change  its  value  as  p  changes  its  value  ;  or, 
P  will  change  with  the  square  root  of  p  :  a  property 
usually  expressed  by 


FOCAL  PERPENDICULAR  ON  TANGENT.        447 

Theorem  XL — The  focal  perpendicular  upon  the  tangent 
of  a  parabola  varies  in  the  subduplicate  ratio  of  the  focal 
radius  of  contact. 

587.  Focal  Perpendicular  in  terms  of  its  inclination  to 
the  Axis — The  perpendicular  from  (p,  0)  upon  the  line  whose 
equation  (Art.  577,  Cor.)  is 

m2x  —  my  -f-  p  =  0, 
will  be,  according  to  Art.  105,  Cor.  2, 

p  _     m*p  +  p      _  p  ^ 

l/(m*+m''!)       m 

Put  0  =  the  inclination  of  P,  measured  from  the  axis  toward  the 
right:  then  m  =  cot 0,  and  we  get 

P=^sec0. 

588.  The  equation  to  the  tangent  being  written  (Art. 
577,  Cor.) 

that  of  its   focal   perpendicular,   which   passes   through 
(p,  0),  will  be 

Combining  these  so  as  to  eliminate  m,  we  get 


as  the  equation  to  the  locus  of  the  point  in  which  the 
focal  perpendicular  meets  the  tangent.  Hence,  (Art. 
580,  Cor.,) 

Theorem  XII— The  locus  of  the  foot  of  the  focal  per- 
pendicular upon  any  tangent  of  a  parabola,  is  the  tangent 
at  the  vertex  of  the  curve. 

Corollary  1. — By  means  of  this  property,  we  can  solve 
in"  its  most  general  form  the  problem. 
An.  Ge.  41. 


448 


ANALYTIC  GEOMETRY. 


\ 


To  draw  a  tangent  to  a  parabola  through  any  given 
point.  —  Let  P  be  the  given 
point,  and  join  it  with  the  focus 
F.  On  PF  as  a  diameter,  de- 
scribe a  circle  cutting  the  ver- 
tical tangent  in  Q  and  Q' :  these 
points  will  be  the  feet  of  focal 
perpendiculars,  as  the  angles 
PQF,  PQ'F  are  inscribed  in 
semicircles.  Hence,  a  line  join- 
ing P  to  either  of  them,  for  instance  the  line  PQ,  will 
touch  the  parabola  in  some  point  T. 

If  the  point  of  contact  is  required,  produce  the  axis  to 
meet  PQ  in  72,  and  then  apply  the  method  of  Art.  582, 
Cor.  When  the  given  point  is  on  the  curve,  as  at  T,  the 
auxiliary  circle  will  touch  the  vertical  tangent ;  but  the 
point  Q  can  still  be  found,  by  dropping  a  perpendicular 
from  the  middle  point  of  FT  upon  A  Y. 

Corollary  2. — The  present  theorem,  and  the  resulting 
construction,  completely  justify  the  view,  taken  in  Art. 
567,  Cor.  1,  that  the  vertical  tangent  of  the  Parabola  is 
the  analogue  of  the  circle  circumscribed  about  the  Ellipse. 
We  thus  arrive  at  the  conclusion,  often  serviceable  in  anal- 
ysis, that  the  Right  Line  may  be  denned  as  the  circle  whose 
radius  is  infinite. 

589.  Since  the  vertical  tangent  is  the  locus  of  the  foot 
of  the  focal  perpendicular  on  any  other  tangent,  that  is, 
the  line  in  which  the  foot  of  every  such  perpendicular  is 
found,  it  follows  that  every  right  line  drawn  from  the 
focus  to  the  vertical  tangent  is  a  focal  perpendicular  to 
some  other  tangent  of  the  curve.  Besides,  it  is  obvious 
that  any  given  point  and  right  line  may  be  regarded  as 
the  focus  and  vertical  tangent  of  some  parabola.  Hence, 


PARABOLA  AS  ENVELOPE.  449 

Theorem  XIII. — If  from  any  point  a  line  be  drawn  to 
a  fixed  right  line,  and  a  perpendicular  to  it  be  formed  at 
the  intersection,  the  perpendicular  will  be  tangent  to  the 
parabola  of  which  the  point  and  fixed  line  are  the  focus 
and  vertical  tangent. 

Corollary — Thus  we  see  that  a  parabola  is 
the  envelope  of  such  a  perpendicular,  and  we 
may  therefore  approximate  the  outline  of  one, 
by  drawing  oblique  lines  to  a  given  right  line 
from  a  fixed  point  P,  and  erecting  perpendicu- 
lars at  their  extremities.  If  formed  close  enough 
together,  these  perpendiculars  will  define  the 
curve  with  considerable  distinctness,  as  the 
diagram  shows. 

59O.  Let  us  now  ascend  from  the  theorem  of  Art.  588 
to  the  general  one,  in  which  a  line  from  the  focus  meets 
the  tangent  at  any  fixed  angle. 

Calling  this  angle  0,  and  writing  the  equation  to  the  tangent 
(Art.  577^  Cor.) 

y=«*+£  (i), 

we  obtain,  for  the  equation  to  the  intersecting  line  from  the  focus 

(Art.  103), 

m  +  tan0    ;,_^  ^ 


From  (2),  by  clearing  of  fractions,  expanding,  and  collecting  terms, 

y  —  mx  •=  (1  +  ™2)  ^  tan  6  —  mp. 

Subtracting  this  result  from  (1)  member  by  member,  and  then 
transposing, 

(1  -f  m2)  x  tan  6  =  \  -  +  mp. 
Dividing  through  by  (1  +  m2), 

P  in 

x  tan  d=—    .-.    m  = 


450 


ANALYTIC  GEOMETRY. 


Substituting  this  value  of  m  in  (1),  we  get  the  equation  to  the  locus 
of  the  intersection  of  the  tangent  and  the  focal  line,  namely, 

y  =  x  tan  6  +  p  cot  0. 

But  (Art.  577,  Cor.)  this  denotes  a  tangent  of  the  same  parabola, 
inclined  to  the  axis  at  the  angle  6.     Hence, 

Theorem  XIV The  locus  of  the  intersection  of  a  tangent  with  the 

focal  line  which  meets  it  at  a  fixed  angle,  is  the  tangent  which  meets 
the  axis  at  the  same  angle. 

591.    Angle  between   any  two  Tangents.— Let  x'y',  x"y" 
denote  the  two  points  of  contact  Q,  Q/ '. 
The  equations  to  the  corresponding  tan- 
gents (Art.  576)  will  then  be 


Hence,  applying  the  formula  of  Art.  96, 
Cor.  1,  we  get 


tan 


592.  This  expression  leads  to  a  noticeable  property  of  the 
Parabola,  as  follows  :—  PQ,  PQf  being  any  two  intersecting  tangents, 
the  equations  to  their  focal  radii  of  contact  FQ,  FQ/  (Art.  95) 

will  be 

y  /  y     _    y" 

x 


p 


x'—p        x—p       x"—p 


Substituting  for  x'  and  re"  their  values  from  the  equation  to  the 
Parabola,  and  clearing  of  fractions,  we  may  write  these  equations, 


(FQ), 


(x-p) 

-  4/)  y  =  4py"  (x  -  p) 


From  them,  by  Art.  96,  Cor.  1,  we  get 

/2  -  •  V)  - 


n 


o2)2  -  4p*  (y"  -  3/02 


NORMAL  OF  THE  PARABOLA.  451 

Now,  if  we  apply  the  formula  for  the  tangent  of  a  double  angle 
(Trig.,  847,  in)  to  the  angle  QPQ',  whose  tangent  we  found  in  the 
preceding  article,  we  shall  get,  for  the  tangent  of  2QPQ',  the  ex- 
pression just  obtained.  Hence  QFQ/  =  2QPQ/;  and  we  have 

Theorem  XV. —  The  angle  between  any  two  tangents  of  a  parabola 
is  equal  to  half  the  focal  angle  subtended  by  their  chord  of  contact. 

593.  The  equations  to  any  two  tangents  of  a  parabola  that  cut 
each  other  at  right  angles  (Arts.  577,  Cor. ;  96,  Cor.  3)  will  be 

,  P  x 

y  =  mx  +  —,     y  =  —      —  mp. 
m  m 

Subtracting  the  second  of  these  from  the  first,  we  obtain 
x  =  —  p 

as  the  equation  to  the  locus  of  the  intersection.  But  this  equation 
denotes  a  right  line  perpendicular  to  the  axis  at  the  distance  p  on 
the  left  of  the  vertex;  in  other  words,  the  line  called  the  directrix 
in  our  primary  definitions.  Hence, 

Theorem  XVI. — The  locus  of  the  intersection  of  tangents  which  cut 
each  other  at  right  angles,  is  the  directrix  of  the  curve. 

Corollary. — Since  this  locus,  in  the  case  of  the  Ellipse,  is  a  circle 
concentric  with  the  curve  (Art.  395),  this  theorem  again  shows  us 
that  the  Circle  converges  to  the  form  of  the  Right  Line  as  its  radius 
tends  to  infinity,  and  that  we  may  therefore  correctly  regard  the 
Right  Line  as  a  circle  with  an  infinite  radius. 

THE    NORMAL. 

594.  Equation  to  the  Normal. — The  equation  to 
the  perpendicular  drawn  through  the  point  of  contact 
x'y'  to  the  tangent 

y'y  =  2p(x  +  O, 
is  found  by  Art.  103,  Cor.  2,  and  is  therefore 


452 


ANALYTIC  GEOMETRY. 


595.  Let  P  be  any  point  x'y'  on  a  parabola,  PN  the 

corresponding  normal,  DP  a  diameter 
through  P,  and  FP  the  focal  radius 
of  its  vertex.  The  equation  to  FP 
(Art.  95)  is 

y'x  —  (xt—p)y=pyf. 

Comparing  this  with  the  equation  to 
the  normal,  we  get  (Art.  96,  Cor.  1) 


1  FPN== 
- 


But,  since  every  diameter  is  parallel  to  the  axis,  DPN= 
,  and  we  have,  from  the  equation  to  the  normal, 


Hence,  FPN=  180°  —  DPN  =  QPN-,  and  we  obtain 

Theorem  XVII. — The  normal  of  a  parabola  bisects  the 
external  angle  between  the  corresponding  diameter  and 
focal  radius. 

Corollary. — To  construct  a  normal  at  any  point  P  of 
the  curve,  we  therefore  draw  the  focal  radius  PF,  and 
the  diameter  DPQ,  laying  off  upon  the  latter  PQ  =  PF, 
and  joining  QF:  then  will  PN,  drawn  perpendicular  to 
QF,  bisect  the  angle  FPQ,  and  for  that  reason  be  the 
normal  required. 

596.  Intercept  of  the  Normal. — Making  y  =  0  in 
the  equation  of  Art.  594,  we  obtain 


x  =  AN=2p  -f  xf. 

Corollary.  —  This  result  enables  us  to 
construct  a  normal  at  any  point  of  the 
curve,  or  from  any  point  on  the  axis. 


SUBNORMAL  OF  THE  PARABOLA.  453 

For,  if  the  point  P  is  given,  we  draw  the  corresponding 
ordinate  PM,  and  lay  off  MN  to  the  right  of  its  foot, 
equal  to  2AF:  we  thus  find  JV,  the  foot  of  the  required 
normal.  When  N  is  given,  we  lay  off  NM—  2AF  to  its 
left,  erect  the  ordinate  MP,  and  join  NP. 

597.  Since  FN=AN—AF=(2p+xr)—p=p  +  x!, 

we  have  (Arts.  568,  582) 

Theorem  XVIII. —  The  foot  of  the  normal  is  at  the  same 
distance  from  the  focus  as  the  foot  and  the  point  of  contact 
of  the  corresponding  tangent. 

Corollary. — This    is    the    same    as 
saying   that   the   three    points    men- 
tioned  are    on  the  same   circle,   de- 
scribed   from    the    focus    as    center. 
Hence,  to   construct  either  the   tan- 
gent or  the  normal,  or  both,  pass  a 
circle    from    the    center    F    through 
either  of  the  three  points  P,  T,  N,  as  one  or  another  is 
given,  and  join  the  points  in  which  it  cuts  the  axis  with 
the  point  in  which  it  cuts  the  parabola. 

598.  L,ength   of  the   Subnormal. — For    this    (see 
diagram,  Art.  596),  we  have  MN=AN—AM.     That 
is  (Art.  596), 

subnor  =  2p. 

In  other  words  (Art.  561,  Cor.),  we  have  obtained 

Theorem  XIX. —  The  subnormal  of  a  parabola  is  constant, 
and  equal  to  twice  the  distance  from  the  focus  to  the  vertex. 

599.  Length    of  the    Normal. — For    the    distance 
between   x'yr   and    (2p  -f  a/,  0),   we    have    (Art.   51,   I, 
Cor.  1) 

PN2  =  4p2  +  y12  = 


454  ANALYTIC  GEOMETRY. 

Now  (Art.  568)  p  +  d  =  p.     Hence, 
PN2  =  4pp. 

But  (Art.  585)  pp  =  the  square  of  the  focal  perpendicu- 
lar on  the  tangent  at  P.     Therefore, 

Theorem  XX. —  The  normal  of  a  parabola  is  double  the 
focal  perpendicular  on  the  corresponding  tangent. 

ii.  THE  CURVE  IN  TERMS  OF  ANY  DIAMETER. 

GOO.  The  equation  which  we  have  thus  far  employed 
is  only  a  special  form  of  a  more  general  one,  and  many 
of  the  properties  proved  by  means  of  it  are  but  particu- 
lar cases  of  generic  theorems  which  relate,  not  to  the 
axis,  but  to  any  diameter  whatever.  The  truth  of  this 
will  appear  as  soon  as  we  transform 
7/2 1=  4pXy  which  is  referred  to  the  axis 
AX  and  the  vertical  tangent  AY.  to 

O 

any   diameter   A'X'   and    its   vertical 

tangent    A'Yf.      This    transformation 

we  c;m  effect  by  means  of  the  formulae 

in  Art.  56,  Cor.  1,  observing  that  the 

new  axis  of  x  is  parallel  to  the  primitive,  and  the  «  of  the 

formulae  therefore  equal  to  zero.     If  in  addition  we  call 

the  angle  Y'TX  not  /?  but  0,  and  put  x'y'  to  denote  the 

new   origin    A1,  these   formulae    of  transformation    will 

become  (Art.  58)  .    , 

x  =  xf  -f  x  -f-  y  cos  0, 

y  =  y'  +  y sin  0- 

6O1.  Equation  to  the  Parabola,  referred  to  any 
Diameter  and  its  Vertical  Tangent. — Replacing  the  y 
and  x  of  -if  =  4px  by  their  values  as  given  in  the  pre- 
ceding formulae,  and  collecting  the  terms,  we  get 

if  sin2  d  +  Z(yr  sin  6  —  2p  cos  0)  y  -f-  ?/'2  —  4pxf  =  4px. 


PARABOLA  REFERRED  TO  ANY  DIAMETER.    455 

But,  as  the  new  origin  x'y1  is  on  the  curve,  y'2  —  4px'—  0. 
Also,  since  the  new  axis  of  y  is  a  tangent,  tan0  =  2p  :  T/'; 
so  that  y'  sin  6  —  2p  cos  6  =  0.  Hence,  the  transformed 
equation  is  in  reality 


if  sin2/?  =  lpx  (1). 

Putting  p  :  sin2  6  =  p',  we  may  write  the  equation 

y2   _.   4^/3. 


This  at  once  shows  that  y2  =  4px  is  the  form  which  (1) 
assumes  when  the  diameter  chosen  for  the  axis  of  x  is 
that  whose  vertical  tangent  is  perpendicular  to  it,  so 
that  sin2  0=1. 

602.  Before  employing  our  new  equation  itself,  we 
may  extend  the  property  of  Theo- 
rem I  by  means  of  relations  derived 

from  y2  =  4px :  this  will  better  pre- 
pare the  way  for  the  use  of  y2  =  4p'x. 

Let  ZM/ be  any  diameter,  and  A!  T  T 
its  vertical  tangent.  Parallel  to  the 
latter,  draw  FF'  through  the  focus. 
Then,  in  the  parallelogram  F'T,  we 
shall  have  A'F'  =  FT  =  (Art.  582)  FA'.  But,  from 
the  definition  of  the  curve,  supposing  IfD  to  be  the 
directrix,  FA'  =  AD.  Hence,  A'D  =  A'F' ;  or, 

Theorem  XXI. —  The  vertex  of  any  diameter  bisects  the 
distance  from  the  directrix  to  the  point  in  which  the 
diameter  is  cut  by  its  focal  ordinate. 

603.  The  factor  p'  =p  :  sin2  0  which  enters  the  second 
member  of  equation  (2),  Art.  601,  may  be  expressed  in 
terms  of  p  and  x',  by  the  following  process : 


456  ANALYTIC  GEOMETRY.       . 

According  to  Art.  576,  tan  6  —  2p  :  y'.     Hence, 

sin  0  =  / 


We  have,  then, 

Now  (Art.  568)  p  -f  x'  =  FAf,  the  focal  distance  of  the 
vertex  of  any  diameter.  Hence, 

Theorem  XXIL— The  focal  distance  of  the  vertex  of  any 
diameter  is  equal  to  the  focal  distance  of  the  principal 
vertex,  divided  by  the  square  of  the  sine  of  the  angle  between 
the  diameter  and  its  vertical  tangent. 

6O4.  The  expression  just  obtained  aids  us  to  interpret 
the  new  equation  y2  =  4p'x.  For,  from  what  precedes, 
the  equation  may  be  written 

?/2  —  4  (p  -f  x')  x, 

and  we  learn  that  the  symbol  p'  signifies  the  focal 
distance  of  the  vertex  taken  for  the  new  origin.  We 
therefore  read  from  the  equation  at  once,  the  following 
extension  of  Theorem  III : 

Theorem  XXIII. —  The  square  on  an  ordinate  to  any 
diameter  is  equal  to  four  times  the  rectangle  under  the 
corresponding  abscissa  and  the  focal  distance  of  the 
vertex. 

Corollary. — Hence,  the  squares  on  the  ordinates  to 
any  diameter  vary  as  the  corresponding  abscissas. 


FOCAL  DOUBLE  ORDINATES.  457 

605.  In  Art.  602,  we  found  A'F'  =  FAr  ^=  p  +  x'. 

Hence,  making  x  =p  -f-  x'  in  the  equa- 
tion of  Art.  604,  we  get 


,p 

bfr- — x 

Now,  since  every  diameter  bisects  the        & 
chords  parallel  to  its  vertical  tangent, 
PQ  =  2F'P=4FA'-,  and  we  have 

Theorem  XXIV. —  The  focal  double  ordinate  to  any 
diameter  is  equal  to  four  times  the  focal  distance  of  its 
vertex. 

Remark. — The  value  of  the  latus  rectum  (Art.  566), 
furnishes  a  particular  case  of  this  theorem ;  and,  as  4p 
signifies  the  length  of  the  focal  double  ordinate  to  the 
axis,  so  4p',  by  what  precedes,  represents  that  of  the 
focal  double  ordinate  to  any  diameter.  From  Arts.  429, 
522,  we  see  that  this  uniform  analogy  among  the  focal 
double  ordinates  to  all  diameters,  is  peculiar  to  the 
Parabola. 

6O6.  The  reader  may  now  interpret  the  equation 


and  show  by  means  of  it,  that,  with  reference  to  any 
diameter,  the  Parabola  consists  of  a  single  infinite 
branch,  tending  to  two  parallel  right  lines  as  its  limiting 
form,  and  symmetric  to  the  diameter. 

DIAMETRAL  PROPERTIES  OF  THE   TANGENT. 

607.  Equation  to  the  Tangent,  referred  to  any 
Diameter. — From  the  identity  of  form  in  the  equations 
y2  =  4p.r,  y2  =  4pfx,  we  at  once  infer  that  this  must  be 


458  ANALYTIC  GEOMETRY. 

GO8.  Making  y  =  0  in  this  equation,  we  get,  for  the 
intercept  of  the  tangent  on  any  diameter, 

x  =  —  x'. 

This  shows  that  the  tangent  cuts  any  diameter  on  the 
left  of  its  vertex,  at  a  distance  equal  to  the  abscissa  of 
contact.  Thus  the  vertex  of  any  diameter  is  situated 
midway  between  the  foot  of  the  tangent  and  that  of  its 
ordinate  of  contact,  and  we  have,  as  the  extension  of 
Theorem  X, 

Theorem  XXV. — The  subtangent  to  any  diameter  of  a 
parabola  is  bisected  in  tlie  corresponding  vertex. 

Corollary  1. — This  property  enables  us  to  construct  a 
tangent  to  a  parabola  from  any 
external  point  whatever.  For,  if 
T'  be  such  point,  we  have  only 
to  draw  the  diameter  TfMr\  form 
the  tangent  A'T  at  its  vertex 
(by  dropping  a  perpendicular 
from  A'  upon  the  axis,  and 
setting  off  AT  =  the  abscissa  thus  determined),  take 
A'M'  =  A'T',  draw  M'P  parallel  to  the  tangent  A'T, 
and  join  PT'. 

Corollary  2. — By  the  same  property,  we  can  construct 
an  ordinate  to  any  diameter.  This  is  either  done  in  the 
way  M'P  was  formed  above,  or  as  follows:  —  Take  any 
point  T"  on  the  axis,  make  AM='AT",  erect  the  per- 
pendicular MP,  and  join  PT".  Then,  T1  being  the 
point  where  this  tangent  cuts  the  given'  diameter,  take 
A'M'  =  A'T',  and  join  M'P. 

6O9.  Let  PQ  be  any  chord  of  a  parabola.  Then 
(Art.  607)  the  equations  to  the  tangents  at  its  opposite 


POLAR  IN  THE  PARABOLA.  459 

extremities  P  and  $,  by  referring  them  to  its  bisecting 
diameter,  will  be 

2p'  (x  +  x')  —  y'y  =  0,      2p'  (x  +  xf)  +  yfy  =  0. 
Subtracting  the  first  of  these  from  the  second,  we  get 


as  the  equation  to  the  locus  of  the  intersection.     Hence, 
Theorem  XXVI.  —  Tangents  at  the   extremities  of  any 
chord  of  a  parabola  meet  on  the  diameter  which  bisects 
that  chord. 

POLE    AND    POLAR. 

610.  We  shall  now  prove  that  the  polar  relation  is  a 
property  of  the  Parabola,  following  the  same  steps  as  in 
the  Ellipse  and  the  Hyperbola. 

611.  Chord  of  Contact  in  the  Parabola.  —  Let  xfy' 
denote  the  point  from  wrhich  the  two  tangents  that  deter- 
mine the  chord  are  drawn,  and  x^l9  x$2  the  extremities 
of  the  chord.     Since  x'yf  is  upon  both  tangents,  we  have 

i/y  -  2p'  (x>  +  x,),      y,y'  =  2/  (x'  +  x2). 

That  is,  the  two  extremities  of  the  chord  are  on  the  line 
whose  equation  is 

y'y  =  2pf  (x  +  xT). 

And  this  is  therefore  the  equation  to  the  chord  itself. 

612.  Locus  of  the  Intersection  of  Tangents  to  the 
Parabola.  —  Let  x'y'  be  the  fixed  point  through  which 
the  chord  of  contact  of  the  intersecting  tangents  is  drawn. 
Then,  if  x^y^  be  the  intersection  of  the  two  tangents,  since 
x'yf  is  always  on  their  chord  of  contact,  we  shall  have 
(Art.  611) 

y,y'  =  Zp'^  +  xJ. 


460  ANALYTIC  GEOMETRY. 

And  this  being  true,  however  x}yl  may  change  its  position 
as  the  chord  of  contact  revolves  about  x'y',  the  co-ordi- 
nates of  intersection  must  always  satisfy  the  equation 


This,  therefore,  is  the  equation  to  the  locus  sought. 

613.  Tangent  and  Chord  of  Contact  taken  iip 
into  the  wider  conception  of  the  Polar.  —  Here,  too, 
as  well  as  in  the  Ellipse  and  the  Hyperbola,  the  two 
equations  just  found  are  identical  in  form  with  that  of 
the  tangent.  By  the  same  reasonmg,  then,  as  in  Arts. 
433,  526,  we  learn  that  the  tangent  and  chord  of  contact 
in  the  Parabola  are  particular  cases  of  the  locus  just  dis- 
cussed. Now,  too,  by  its  equation,  this  locus  is  a  right 
line;  and,  if  we  suppose  x'y'  to  be  any  point  on  a  given 
right  line,  the  co-efficients  of  the  equation  in  Art.  611 
will  fulfill  the  condition  Ax'  -}-  By'  +  C=  0,  and  thus 
(Art.  117)  the  chord  of  contact  will  pass  through  a  fixed 
point.  In  the  curve  now  before  us,  therefore,  we  have 
the  twofold  theorem  : 

I.  If  from  a  fixed  point  chords  be  drawn  to  any  para- 
bola, and  tangents  to  the  curve  be  formed  at  the  extremities 
of  each  chord,  the  intersections  of  the  several  pairs  of  tan- 
gents will  lie  on  one  right  line. 

II.  If  from  different  points  lying  on  one  right  line  pairs 
of  tangents  be  drawn  to  any  parabola,  their  several  chords 
of  contact  will  meet  in  one  point. 

Thus  the  law  that  renders  the  locus  of  Art.  612  the 
generic  form  of  which  the  tangent  and  chord  of  contact 
are  special  phases,  is  the  law  of  polar  reciprocity:  whence 
the  Parabola,  in  common  with  the  other  two  Conies,  im- 
parts to  every  point  in  its  plane  the  power  of  determining 
a  right  line;  and  reciprocally. 


POLAR  IN  THE  PARABOLA.  461 

614.  Equation  to  the  Polar  with  respect  to  a 
Parabola.  —  This,  as  we  gather  immediately  from  the 
preceding  results,  is 


if  referred  to  any  diameter  ;  or,  if  referred  to  the  axis, 


615.  Definitions.  —  The  Polar  of  any  point,  with  re- 
spect to  a  parabola,  is  the  right  line  which  forms  the 
locus  of  the  intersection  of  the  two  tangents  drawn  at 
the  extremities  of  any  chord  passing  through  the  point. 

The  Pole  of  any  right  line,  with  respect  to  the  same 
curve,  is  the  point  in  which  all  the  chords  of  contact 
corresponding  to  different  points  on  the  line  intersect. 

We   have,  then,  the  following   constructions:  —  When 
the  pole  P  is  given,  draw  through  it 
any  two  chords  T'T,  S'S,  and  form 
the  corresponding  pairs  of  tangents, 
T'L  and  TL,  S'M  and  SM:  the  line 
LM  which  joins  the  intersection  of     M 
the  first  pair  to  that  of  the  second, 
will  be  the  polar  of  P.     When   the 
polar  is  given,  take  any  two  of  its 
points,  as  L  and  M,  draw  a  pair  of  tangents  from  each, 
and  form  the  corresponding  chords  of  contact,  T'T,  S'S: 
the  point  P  in  which  these  intersect,  will  be  the  pole  of 
LM. 

When  the  pole  is  without  the  curve,  as  at  M9  the  polar 
is  the  corresponding  chord  of  contact  $'S'9  and  when  it 
is  on  the  curve,  as  at  T,  the  polar  is  the  tangent  at  T. 
In  these  cases,  the  drawing  may  be  made  in  accordance 
with  the  facts. 


462  ANALYTIC  GEOMETRY. 

616o  Direction    of  the    Polar.  —  By   referring  the 

polar  of  any  point  to  the  diameter  drawn  through  the 

point,  the  y'  of  its  equation  will  become  =  0,  and  the 
equation  itself  (Art.  614)  will  assume  the  form 

x  =  —  xf. 

This  denotes  a  parallel  to  the  axis  of  y.     Hence, 

Theorem  XXVII.  —  The  polar  of  any  point,  with  respect 
to  a  parabola,  is  parallel  to  the  ordinates  of  the  diameter 
which  passes  through  the  point. 

Corollary  1.  —  The  equation  x  =  —  xr,  more  exactly 
interpreted,  gives  us  :  The  polar  of  any  point  on  a 
diameter  is  parallel  to  the  ordinates  of  the  diameter,  and 
its  distance  from  the  vertex  of  the  diameter  is  equal,  in  an 
opposite  direction,  to  the  distance  of  the  point.  And,  in 
particular,  The  polar  of  any  point  on  the  axis  is  the  per- 
pendicular which  cuts  the  axis  at  the  same  distance  from 
the  vertex  as  the  point  itself,  but  on  the  opposite  side. 

Corollary  2.  —  To  construct  the  polar,  therefore,  draw 
a  diameter  through  the  pole,  take  on  the  opposite  side 
of  its  vertex  a  point  equidistant  with  the  pole,  and  draw 
through  this  a  parallel  to  the  corresponding  ordinates. 

617.  Polar  of  the  Focus.  —  The  equation  to  this  is 
obtained  by  putting  (p,  0)  for  x'y'  in  the  second  equation 
of  Art.  614,  and  is 


The  focal  polar  of  the  Parabola  is  therefore  identical 
with  the  line  which  in  Art.  180  we  named  the  directrix, 
and  we  shall  presently  see  that  our  ability  to  generate 
the  curve  by  the  means  employed  in  Art.  179,  is  due  to 
the  polar  relation  of  that  line  to  the  focus. 


POLAR  OF  THE  FOCUS. 


463 


G18.  Let  D'D  represent  the  polar  of  the  focus  F. 
Then,  obviously,  PD  =  RA  +  AM  ; 
or,  the  distance  of  any  point  on  the 
curve  from  the  polar  is  equal  to  the 
distance  of  the  polar  from  the  vertex, 
increased  by  the  abscissa  of  the  point. 
That  is, 

PD  =     +  x. 


=  PJ)'}  or, 


Now  (Art.  568)  p  +  x  =  FP\  whence, 
since  e  =  1  in  the  Parabola, 

FP 


Here,  then,  the  property  of  Arts.  439,  532  again  appears, 
and  we  have 

Theorem  XXVIII,  —  The  distance  of  any  point  on  a 

parabola  from  the  focus  is  in  a  constant  ratio  to   its 

distance   from   the  polar   of  the  focus,   the   ratio  being 
equal  to  the  eccentricity  of  the  curve. 

Remark  1,  —  We  thus  complete  the  circuit  of  our  anal- 
ysis, and,  as  stated  above,  return  upon  the  property  from 
which  we  set  forth  in  Art.  179.  The  significance  of  our 
present  result  consists  in  the  fact,  that  we  have  translated 
the  apparently  arbitrary  definition  of  Art.  179  into  the 
generic  law  of  polarity.  And  we  may  say  that  we  have 
vindicated  our  method  of  generating  the  curve  ;  because 
we  can  now  see  that  it  is  the  mechanical  expression  of 
the  power  to  determine  a  conic,  which  the  Point  and  the 
Right  Line  together  possess  :  a  power  reciprocally  in- 
volved, of  course,  in  that  of  the  Conic  to  bring  these  two 
Forms  into  the  polar  relation. 
An.  Ge.  42. 


464  ANALYTIC  GEOMETRY. 

It  may  deserve  mention,  that  the  construction  of  Art. 
179,  like  those  of  the  corollaries  to  Arts.  439,  532,  em- 
ploys the  parts  of  a  right  triangle  in  order  to  embody 
the  constant  ratio  of  the  focal  and  polar  distances,  the 
constant  in  the  case  of  the  Parabola  being  the  ratio  of 
the  base  to  itself. 

Remark  2.  —  The  name  parabola  thus  acquires  a  new 
meaning.  We  may  henceforth  regard  it  as  signifying 
the  conic  in  which  the  constant  ratio  between  the  focal  and 
polar  distances  equals  unity. 

G1O.    Focal    Angle   subtemlcd    by   any    Tangent.  —  By    an 

analysis  similar  to  that  of  Art.  440,  x  being 
the  abscissa  of  the  point  P  from  which  the 
tangent  is  drawn,  p  its  radius  vector  FP, 
and  £>  the  angle  PFT,  we  can  show  that  P< 


62O.  This  expression,  like  those  of  Arts.  440,  533,  is  inde- 
pendent of  the  point  of  contact.  Hence,  the  angle  PFT  =  the 
angle  PFT'  ;  and,  with  respect  to  the  whole  angle  TFT',  we  get 

Theorem  XXIX.  —  The  right  line  that  joins  the  focus  to  the  pole  of 
any  chord,  bisects  the  focal  angle  which  the  chord  subtends. 

Corollary.  —  In  particular,  The  line  that  joins  the  focus  to  the  pole 
of  any  focal  chord  is  perpendicular  to  the  chord. 

Remark.  —  By  comparing  this  corollary  with  the  theorem  of  Art. 
593,  and  bearing  in  mind  that  the  directrix,  as  the  polar  of  the 
focus,  is  the  line  in  which  the  pair  of  tangents  drawn  at  the  ex- 
tremities of  any  focal  chord  will  intersect,  we  may  state  the  follow- 
ing noticeable  group  of  related  properties  : 

If  tangents  be  drawn  at  the  extremities  of  any  focal  chord  of  a 
parabola, 

I  .    The  tangents  will  intersect  on  the  directrix. 

2.  The  tangents  will  meet  each  other  at  right  angles. 

3.  The  line  that  joins  their  intersection  to  the  focus  will  be  perpen- 
dicular to  the  focal  chord. 


PARAMETER  OF  THE  PARABOLA.  465 

621.  We  shall  in  this  article  solve  two  examples, 
which  will  show  the  beginner  IIOAV  to  take  advantage  of 
such  results  as  we  have  lately  obtained. 

I.  Given  two  points  P,  Q,  and  their  polars  T'T,  8'iS:  to  deter- 
mine the  relation  between  the  intercept  wm,  cut  off  on  the  axis  by 
the  polars,  and  the  intercept  MN,  cut  off  by  perpendiculars  from 
the  points. 

Let  x'y'y  x"y"  denote  P  and  Q.     Then 


Prt 


MN=x"  -x'. 
But  the  equations  to  the  polars  T'  T,  S'S  are 

y'y-=  2/>(*  +  »'),     y"y=2i>(x  +  *"); 

and  if  we   make  y  —  0  in  these,  and   take   the 
difference  of  the  results, 


Hence,  The  intercept  on  the  axis  between  any  tico  2^olars  is  equal  to  that 
between  the  perpendiculars  from  their  2>olcs. 

II.  To  prove  that  the  circle  which  circumscribes  the  triangle 
formed  by  any  three  tangents,  passes  through  the  focus. 

Let  L,  Q,  R  be  the  intersections  of  the  tangents,  and  F  the  focus. 
By  Art.  592,  the  angle  L  QR  is  half  the  focal  angle  subtended  by  S'S. 
Also,  by  Art,  620,  the  angle  LFR  =  LFT  +  TFR=18Q°  minus  half  this 
same  focal  angle.  Hence,  LQR  +  LFR  =  180°,  and  the  quadrilateral 
LQRF  is  an  inscribed  quadrilateral.  That  is,  F  is  on  the  circle  which 
circumscribes  the  triangle  LQR. 

PARAMETERS. 

622.  Definition.  —  The    Parameter    of   a    parabola, 
with  respect  to  any  diameter,  is  a  third  proportional  to 
any   abscissa   formed   on   the  diameter,   and   the  corre- 
sponding ordinate.     Thus, 

?/2 
parameter  =  *~-  . 

3/ 

623.  From  the  equation  to  the  Parabola,  y'2  :  xf  =  4p'. 
Also,  from  Art.  604,  pf  =  the  distance  of  the  vertex  of 
a  diameter  from  the  focus.     Hence, 


466  ANALYTIC  GEOMETRY. 

Theorem  XXX.  —  The  parameter  of  any  diameter  in  a 
parabola  is  equal  to  four  times  the  focal  distance  of  its 
vertex. 

Corollary.  —  The  parameter  of  the  axis,  or,  as  it  is 
usually  called,  the  principal  parameter,  is  therefore 
equal  to  four  times  the  distance  from  the  focus  to  the 
vertex  of  the  curve  ;  that  is,  its  value  is  4p. 

Remark.  —  In  the  equations  y2  =  4p'x,  y2  =  4px,  we 
are  henceforward  to  understand  that  p1  is  one-fourth 
the  parameter  of  the  reference-diameter,  and  p  one- 
fourth  the  parameter  of  the  axis.  Or,  with  greater 
generality,  in  any  equation  of  the  form 


P,  the  constant  co-efficient  of  x,  is  to  be  interpreted 
as  the  parameter  of  the  corresponding  parabola,  taken 
with  respect  to  the  diameter  to  which  the  equation  is 
referred. 

624.  In  Art.  605,  we  proved  that  the  focal  double 
ordinale  to  any  diameter  is  equal  to  four  times  the  focal 
distance  of  its  vertex.  Hence, 

Theorem  XXXI.  —  The  parameter  of  any  diameter  in 
a  parabola  is  equal  to  the  focal  double  ordinate  of  that 
diameter. 

Corollary.  —  Accordingly,  the  parameter  of  the  axis  is 
equal  to  the  latus  rectum  :  as  we  might  also  infer  from 
the  fact  that  both  are  equal  to  4p. 

Remark.  —  From  Arts.  429,  522,  as  already  noticed  in 
another  connection,  we  see  that  the  Parabola  is  the  only 
conic  in  which  the  parameter  of  every  diameter  is  equal 
to  the  corresponding  focal  double  ordinate. 


PARABOLA  REFERRED  TO  ITS  FOCUS.         467 

625.  It  is  sometimes  useful  to  express  the  parameter 
of  any  diameter  in  terms  that  refer  to  the  axis  of  the 
curve  as  the  axis  of  x.  To  effect  this,  we  either  use  the 

relation 

p'=p  +  d  (1), 

where  x'  is  the  abscissa  of  the  vertex  of  the  diameter 
whose  parameter  is  sought,  measured  on  the  axis ;  or 
else 


where  6  is  the  angle  made  with  the  axis  by  the  tangent 
at  the  vertex  of  the  diameter. 

626.  The  latter  of  the  foregoing  relations  may  be 
interpreted  as 

Theorem  XXXII.  —  The  parameter  of  any  diameter 
varies  inversely  as  the  square  of  the  sine  of  the  angle 
which  the  corresponding  vertical  tangent  makes  with  the 
axis. 

in.  THE  CURVE  REFERRED  TO  ITS  Focus. 

627.  The  polar  equations  to  the  Parabola  (Art.  183 
cf.  Rem.)  being 


P  ''  ~ 


1  ±:  cos  6  ' 

our  present  knowledge  leads  us  to  assign  to  p  its  proper 
meaning,  and  to  describe  the  numerator  in  the  value  of  p 
as  half  the  parameter  of  the  curve. 

Moreover,  since  e  =  1  in  the  Parabola,  we  may  write 

2p 

P  ''  Z  1  ±  e  cos  6  ' 


468  ANALYTIC   GEOMETRY. 

thus  completely  exhibiting  the  analogy  of  these  ex- 
pressions to  the  corresponding  elliptic  equations  of  Art. 
443  :  an  analogy  which  we  partially  established  in  Art. 
184,  and  which  verifies  the  proposition  of  Art.  567,  that 
a  parabola  may  be  regarded  as  an  ellipse  in  which  the 
eccentricity  has  passed  to  the  limiting  value  =  1. 

G2S.  IPolar  Equation  to  the  Tangent. — In  seeking 
this,  we  shall  avail  ourselves  of  the  property  just  men- 
tioned. 

If  in  the  equation  of  Art.  444  we  replace  a  (1  —  e2) 
by  its  value  2p,  as  found  in  Art.  443,  we  may  write  the 
polar  equation  to  an  elliptic  tangent 


cos  (0  —  V)  —  e  cos  0 

Making  e  =  1  in  this,  we  get  the  equation  to  a  parabolic 
tangent,  namely, 


cos  (0  —  0')  —  cos  6 

iv.  AREA  OF  THE  PARABOLA. 

629.  The  area  of  any  parabolic  segment,  included 
between  the  curve  and  any  double  ordinate  to  the  axis, 
may  be  computed  as  follows : 

Supposing  A  -  PMQ  to  be  the  segment 
whose  area  is  sought,  divide  its  abscissa 
AM  into  any  number  of  equal  parts  at 
B,  (7,  D,  .  .  . ,  erect  ordinates  BL,  CN, 
DR,  ...  at  the  points  of  division,  and 
through  their  extremities  Z,  N,  R,  .  .  . 
draw  parallels  to  the  axis,  producing  both 
the  ordinates  and  the  parallels  until  they  meet  as  in  the 


AREA  OF  THE  PARABOLA.  469 

figure.  The  curve  divides  the  circumscribed  rectangle 
UM  into  two  segments  :  and,  by  the  process  just  de- 
scribed, there  will  be  formed  in  both  of  these  a  number 
of  smaller  rectangles,  corresponding  two  and  two  ;  as 
UR  to  EM,  ON  to  ND,  EL  to  LC,  etc. 

Let  x'y',  x'fy''  be  any  two  successive  points  thus  formed 
upon  the  curve  ;  for  instance,  L,  N.     Then 


area  LC  =  y'  (x"  —  x'}  = 
area  EL  =  xr  (y"  -  y'}  = 


To  express,  then,  the  ratio  of  any  interior  rectangle  to 
its  corresponding  exterior  one,  we  shall  have  an  equation 
of  the  form 

LG_  _  y"  +  y'  _  i  |  y"_ 

EL'         y'  f  y'  ' 

Hence,  the  limiting  value  to  which  this  ratio  tends  as  y" 
converges  to  yr,  is  evidently  =  2  :  and  therefore  the 
ratio  borne  by  the  sum  of  the  interior  rectangles  to  the 
sum  of  the  exterior,  also  tends  to  the  limit  2  as  yfr  con- 
verges to  y'.  Now  the  condition  that  y"  may  converge 
to  y'  is,  that  the  subdivision  of  AM  shall  be  continued 
ad  infinitum:  and  if  this  takes  place,  the  sum  o'f  the  in- 
terior rectangles  will  converge  to  the  area  of  the  interior 
segment  APM\  and  the  sum  of  the  exterior,  to  the  area 
of  the  exterior  segment  APU.  Therefore,  APM  = 
2APU;  or,  putting  x  =  AM,  y  =  MP,  and  A  —  the 
area  of  the  interior  segment, 


470  ANALYTIC  GEOMETRY. 

2 
Similarly,  the  segment  AQM=-xy:  whence, 

o 

Theorem  XXXIII. —  The  area  of  a  parabolic  segment 
cut  off  by  any  double  ordinate  to  the  axis,  is  equal  to 
two-thirds  of  the  circumscribing  rectangle. 

Corollary. — Since  the  same  reasoning  is  obviously 
applicable  to  the  equation  y2  =  4p'rc,  we  may  at  once 
state  the  generic  theorem :  The  area  of  a  parabolic  seg- 
ment cut  off  by  a  double  ordinate  to  any  diameter ',  is  equal 
to  two-thirds  of  the  circumscribing  parallelogram. 

EXAMPLES  ON  THE  PARABOLA. 

1.  The  extremities  of  any  chord  of  a  parabola  being  x'y',  x"y", 
and  the  abscissa  of  its  intersection  with  the  axis  being  Z,  to  prove 

that 

xV  =  x1,       y'y"  —  —  4px. 

2.  Two  tangents  of  a  parabola  meet  the  curve  in  x/y/  and  x"y"  \ 
their  point  of  intersection  being  xy,  show  that 


3.  The  area  of  the  triangle  formed  by  three  tangents  of  a  para- 
bola is  half  that  of  the  triangle  formed  by  joining  their  points  of 
contact. 

4.  To  prove  that  the  area  of  the  triangle  included  between  the 
tangents  to  the  parabolas 

y1  —  mx,       yl  —  nx 

at  points  whose  common  abscissa  —  a,  and  the  portion  of  the  cor- 
responding ordinate  intercepted  between  the  two  curves,  is  equal  to 


5.  To  prove  that  the  three  altitudes  of  any  triangle  circum- 
scribed about  a  parabola,  meet  in  one  point  on  the  directrix. 


EXAMPLES  ON  THE  PARABOLA.  471 

6.  The  cotangents  of  the  inclinations  of  three  parabolic  tangents 
are  in  arithmetical  progression,  the  common  difference  being  =•  J: 
prove  that,  in  the  triangle  inclosed  by  these  tangents,  we  shall  have 

area  =  p'2<3 3. 

7.  Given  the  outline  of  a  parabola:  to  construct  the  axis  r.r.d 
the  focus. 

8.  Find  the  equation  to  the  normal  of  a  parabola,  in  terms  ot' 
its  inclination  to  the  axis;  and  prove  that  the  locus  of  the  foot  of 
the  focal  perpendicular  upon  the  normal  is  a  second  parabola,  whose 
vertex  is  the  focus  of  the  given  one,  and  whose  parameter  is  one- 
fourth  as  great  as  that  of  the  given  one. 

9.  Show  that  the  locus  of  the  intersection  of  parabolic  normals 
which  cut  at  right  angles,  is  a  parabola  whose  parameter  is  one- 
fourth  that  of  the  given  one,  and  whose  vertex  is  at  a  distance  =  3p 
from  the  given  vertex. 

10.  The  centers  of  a  series  of  circles  which  pass  through  the 
focus  of  a  parabola,  are  situated  on  the  curve:    prove  that  each 
circle  touches  the  directrix. 

11.  To  find  the  area  of  the  rectangle  included  by  the  tangent 
arid  normal  at  any  point  P  of  a  parabola,  and  their  respective  focal 
perpendiculars ;  and  to  determine  the  position  of  P  when  the  rect- 
angle is  a  square. 

12.  If  two  parabolic  tangents  are  intersected  by  a  third,  parallel 
to  their  chord  of  contact,  the  distances  from  their  point  of  intersec- 
tion to  their  respective  points  of  contact  are  bisected  by  the  third 
tangent. 

13.  Show  that  the  locus  of  the  vertex  of  a  parabola  which  has 
a  given  focus,  and  touches  a  given  right  line,  is  a  circle,  of  which 
the  perpendicular  from  the  given  focus  to  the  given  line  is  a  diameter. 

14.  Show  that,  if  a  parabola  have  a  given  vertex,  and  touch  a 
given  right  line,  its  focus  will  move  along  another  parabola,  whose 
axis  passes  through  the  given  vertex  at  right  angles  to  the  given 
line,  and  whose  parameter  =  the  distance  from  the  given  vertex 
to  the  given  line. 

15.  TP  and  TQ  are  tangent  to  a  given  parabola  at  P  and  Q, 
and  TIFjtfins  their  intersection  to  the  focus:  prove  that 

FP.  FQ  =  FT\ 
An.  Ge.  43. 


472  ANALYTIC  GEOMETRY. 

16.  A  right  angle   moves  in  such  a  manner  that  its  sides  are 
respectively   tangent   to  two   confocal   parabolas    whose    axes    are 
coincident:  to  find  the  locus  of  its  vertex. 

17.  Prove  that  the  pole  of  the  normal  which  passes  through  one 
extremity  of  the  latus  rectum  of  a  parabola,  is  situated  on  the  diam- 
eter which  passes  through  the  other  extremity,  and  find  its  exact 
position  on  that  line. 

18.  The  triangle  included  by  two  parabolic  tangents  and  their 
chord  of  contact  being  of  a  given  area  =  a2,  prove  that  the  locus 
of  the  pole  is  a  parabola  whose  equation  is 


19.  Prove  that,  if  two  parabolas  whose  axes  are  mutually  per- 
pendicular intersect  in  four  points,  the  four  points  lie  on  a  circle. 

20.  If  two  parabolas,  having  a  common  vertex,  and  axes  at  right 
angles  to  each  other,  intersect  in  the  point  x'y':  then,  £  denoting 
the  latus  rectum  of  the  one,  and  I'  that  of  the  other, 

I  :  «'  :  :  /  :   V. 

[This  property  has  a  special  interest,  on  account  of  its  connection  with 
the  ancient  problem  of  the  Duplication  of  the  Cube.  It  affords,  as  the 
reader  will  observe,  a  method  of  determining  graphically  two  geometric 
means  between  two  given  lines  ;  and  was  proposed  for  this  purpose  by 
MENECHMUS,  a  geometer  of  the  school  of  Plato,  in  connection  with  his 
attempt  to  solve  the  problem  just  mentioned.  The  graphic  problem  of 
"  two  means  "  received  a  variety  of  solutions  at  the  hands  of  the  Greek 
geometers,  two  .of  the  most  celebrated  being  discovered  by  DIOCLES  and 
NICOMEDES.  They  are  effected,  respectively,  with  the  help  of  the  curves 
called  the  Cissoid  and  the  Conchoid.'} 


CHAPTER   SIXTH. 

THE   CONIC  IN  GENERAL. 

63O.  Having  in  the  previous  Chapters  become  familiar 
with  the  properties  of  the  several  conies  considered  as 


THE  CONIC  IN  GENERAL.  473 

separate  curves,  let  us  now  ascend  to  the  wholly  generic 
point  of  view,  from  which  we  may  comprehend  them  not 
as  isolated  Forms,  but  as  members  of  a  united  System, 
and,  in  fact,  as  successive  phases  of  a  generic  locus  which 
may  be  called  the  CONIC,  whose  idea  we  sketched  in  the 
eighth  Section  of  Part  I. 

We  may  begin  by  showing  in  what  sense  this  name  is 
descriptive  of  the  system ;  or,  how  the  curves  may  be 
grouped  together  as  sections  of  a  cone. 

THE  THREE  CURVES  AS  SECTIONS  OF  THE  CONE. 

631.  Definitions. — A  Cone  is  a  surface  generated  by 
moving  a  right  line  which  is  pivoted  upon  a  fixed  point, 
along  the  outline  of  any  given  curve    whose  plane  does 
not  contain  the  fixed  point. 

The  fixed  point  is  called  the  vertex  of  the  cone;  the 
given  curve,  its  directrix;  and  the  moving  right  line,  its 
generatrix. 

Since  the  generatrix  extends  indefinitely  on  both  sides 
of  the  vertex,  the  cone  will  consist  of  two  exactly  similar 
portions,  extending  from  the  vertex  in  opposite  directions 
to  infinity.  Of  these,  one  is  called  the  upper  nappe  of  the 
cone;  and  the  other,  the  lower  nappe. 

Any  single  position  of  the  generatrix  is  called  an  ele- 
ment of  the  cone. 

When  the  directrix  is  a  circle,  the  cone  is  called  cir- 
cular; and  the  right  line  drawn  through  the  vertex  and 
the  center  of  the  directrix,  is  termed  the  axis  of  the  cone. 

632.  Definitions. — A  Right  circular  Cone  is  a  cone 
whose  directrix  is  a  circle,  and  whose  axis  is  perpendic- 
ular to  the  plane  of  its  directrix. 

The  section  formed  with  any  cone  by  a  plane,  is 
termed  a  base  of  the  cone ;  consequently,  the  directrix 


474 


ANALYTIC  GEOMETRY. 


of  a  right  circular  cone  may  be  called  its  base.  For 
this  reason,  such  a  cone  is  often  named  a  right  cone  on 
a  circular  base.  The  diagram  of  the  next  article  presents 
an  example  of  one. 

633.  We  shall  prove,  in  the  proper  place  in  Book 
Second,  the  following  propositions,  which  we  ask  the 
student  to  take  upon  trust  for  the  present,  in  order  that 
we  may  use  them  in  grouping  the  three  curves  according 
to  their  geometric  order : 

I.  Every  section  formed  by  passing  a  plane  through 
a  right  circular  cone  is  a  curve  of  the  Second  order. 

II.  If  the  angle  which  the  secant  plane  makes  with 
the  base  is  less  than  that  made  by  the  generatrix,  the 
section  is  an  ellipse. 

III.  If  the  angle  which  the  secant  plane  makes  with 
the  base  is  equal  to  that  made  by  the  generatrix,  the 
section  is  a  parabola. 

IV.  If  the  angle  which  the  secant  plane  makes  with 
the  base  is  greater  than  that  made  by  the  generatrix, 
the  section  is  an  hyperbola. 

These  three  cases  are  represented 
in  the  diagram :  that  of  the  Ellipse, 
at  AE'j  that  of  the  Parabola,  at 
LPR\  and  that  of  the  Hyperbola, 
at  HA-A'H'.  It  is  manifest,  how- 
ever, from  the  fact  that  the  secant 
plane  makes  with  the  base  an  angle 
successively  less  than,  equal  to,  and 
greater  than  the  angle  made  by  the 
generatrix  (whose  angle  must  have 
a  fixed  value  for  any  given  cone), 
that  the  three  sections  may  be  formed  by  a  single 


SECTIONS  OF  A  CONE.  475 

plane,  by  simply  revolving  it  on  the  line  in  which  it 
cuts  the  base.  Beginning  with  it  in  such  a  position 
that  its  inclination  to  the  base  is  less  than  that  of  the 
generatrix  (whose  inclination  is  often  called  the  incli- 
nation of  the  side  of  the  cone),  and  revolving  it  upward 
toward  the  position  of  parallelism  to  the  side,  we  shall 
cut  out  a  series  of  ellipses  of  greater  and  greater 
eccentricity.  When  the  secant  plane  becomes  parallel 
to  the  side,  the  section  will  be  a  parabola.  When  it  is 
pushed  still  farther  upward,  so  that  its  angle  with  the 
base  becomes  greater  than  that  of  the  side,  it  will  reach 
across  the  space  between  the  two  nappes  of  the  cone, 
and  pierce  the  upper  as  well  as  the  lower  one,  and  the 
section  will  be  an  hyperbola,  whose  two  branches  will 
lie  in  the  two  nappes  respectively. 

From  this  it  appears,  that,  granting  the  proposition 
that  the  sections  are  the  curves  mentioned,  the  natural 
geometric  order  in  which  they  occur  is :  Ellipse,  Para- 
bola, Hyperbola.  This  is  the  same  as  their  analytic 
order,  as  we  found  it  in  Art.  200.  We  shall  be  able  to 
give  a  more  explicit  account  of  their  appearance  as  suc- 
cessive phases  of  the  Conic,  so  soon  as  we  have  presented 
a  fuller  view  of  the  modes  in  which  we  can  represent 
that  generic  locus  by  analytic  symbols. 


VARIOUS  FORMS  OF  THE  EQUATION  TO  THE  CONIC. 

634.  Equation  in  Rectangular  Co-ordinates  at 
the  Vertex. — We  have  not  as  yet  referred  the  three 
Conies  to  the  same  axes  and  origin :  let  us  now  do  so, 
by  transforming  the  central  equations  of  the  Ellipse  and 
the  Hyperbola  to  such  a  vertex  of  each  curve  as  will 
correspond  to  the  vertex  of  the  Parabola. 


476  ANALYTIC  GEOMETRY. 

This  will  require  us  to  transform  the  equation 

2/2=^(«2-*2)  (1) 

to  the  left-hand  vertex  of  the  Ellipse  ;  and  the  equation 

2/2=5(*2-«2)  (2) 

to  the  right-hand  vertex  of  the  Hyperbola.  Accordingly, 
putting  x  —  a  for  x  in  (1),  and  x  +  a  for  x  in  (2),  and 
expanding,  we  get 

2b2          b2    ,  2b2      ,    b2 

y2  =  —  •  x  --  -  z2       if  =  —  x  A  --  ,  x2. 
a  a2  a  a2 


Hence,  (Arts.  428,  521,)  the  equations  to  the  Ellipse, 
the  Hyperbola,  and  the  Parabola  may  be  written 


Remembering,  now,  that  b2  :  a2  =  ±  (1  —  e2),  and  that  its 
value  in  the  case  of  the  Parabola  must  therefore  be  =  0, 
we  learn  that  the  equation  to  the  Conic  in  General  is 


in  which  P  is  the  parameter  of  the  curve,  and  R  the  ratio 
between  the  squares  of  the  semi-axes;  and  we  have  the 
specific  conditions 

R  <  0  .  •  .  Ellipse, 
E  =  0  .'.  Parabola, 
R  >  0  .  *  .  Hyperbola. 

Corollary.  —  By  the  three  equations  from  which  y2  = 
Px  +  Ex2  was  generalized,  we  see  that  in  the  Ellipse, 


NAMES  OF  THE  SECTIONS.  477 

the  square  on  the  ordinate  is  less  than  the  rectangle 
under  the  abscissa  and  parameter;  that  in  the  Parabola, 
the  square  is  equal  to  the  rectangle;  and  that  in  the 
Hyperbola,  the  square  is  greater  than  the  rectangle. 

Remark  —  According  to  PAPPUS  (Math.  Coll.,  VII:  c.  A.  D.  350), 
the  names  of  the  three  curves  were  originally  given  to  designate 
this  property.  But  EUTOCIUS  (A.  D.  560)  says  that  the  names  were 
derived  from  the  fact,  that,  according  to  the  ancient  Greek  geome- 
ters, the  three  sections  were  cut  respectively  from  an  acute-angled, 
a  right-angled,  and  an.  obtuse-angled  cone,  by  means  of  a  plane 
always  passing  at  right  angles  to  the  side.  Thus,  if  the  angle 
under  the  vertex  of  the  cone  were  acute,  the  sum  of  that  and  the 
right  angle  made  by  the  secant  plane  with  the  side  would  be  less 
than  two  right  angles,  and  the  name  ellipse  was  given,  either  to 
indicate  this  deficiency,  or  to  show  that  the  curve  would  then  fall 
short  of  the  upper  nappe  of  the  cone.  But  if  the  angle  under  the 
vertex  were  right,  the  mentioned  sum  of  angles  would  be  equal  to 
two  right  angles,  and  the  plane  of  the  curve  consequently  be  par- 
allel to  the  side  of  the  cone  :  to  denote  which  facts,  the  name  para- 
bola was  given.  Finally,  if  the  angle  under  the  vertex  were  obtuse, 
the  mentioned  sum  of  angles  would  be  greater  than  two  right  angles, 
and  the  name  hyperbola  was  given,  either  to  suggest  this  excess,  or 
to  indicate  that  the  curve  would  then  reach  over  to  the  upper  nappe 
of  the  cone. 

It  is  noticeable  here,  that  the  early  geometers  supposed  the  three 
sections  to  be  peculiar  respectively  to  an  acute-angled,  a  right-angled, 
and  an  obtuse-angled  cone.  The  improvement  of  forming  them  all 
from  the  same  cone  by  merely  changing  the  inclination  of  the  secant 
plane,  was  introduced  by  APOLLONIUS  OF  PERGA,  B.  c.  250. 

Which  of  these  etymologies  should  have  the  preference,  is  a 
question  among  critics  of  mathematical  history.  It  is  remarkable, 
however,  that  the  names  represent  equally  well  all  the  distinguish- 
ing properties  of  the  curves,  whether  geometric  or  analytic  :  as  the 
reader  may  perhaps  have  already  observed  for  himself. 

The  name  parameter,  which  simply  means  corresponding  measure, 
or  co-efficient,  is  given  to  the  quantities  '2lr  :  a  and  y''z  :  a/,  on  account 
of  the  position  they  occupy  in  the  equation 


one  or  the  other  of  them  being  the  firs*,  co-efficient  in  the  second 


478  ANALYTIC  GEOMETRY. 

member,  according  as  the  conic  is  central  or  non-central.  More- 
over, as  the  dimensions  of  the  curve  depend  mainly  upon  the  value 
of  P,  the  ratios  which  it  symbolizes  are  naturally  termed  par  excel- 
lence the  measures  of  the  Conic.  On  the  other  hand,  the  name 
parameter  may  be  defined  in  each  conic  by  the  value  which  it 
denotes  for  each,  as  in  the  preceding  Chapters. 

6&5.  Equation  in  terms  of  the  Focus  and  its 
Polar.  —  We  have  seen  (Arts.  439,  532,  618)  that  in 
every  conic  the  distance  of  any  point  on  the  curve  from 
the  focus  is  in  a  constant  ratio  to  its  distance  from  the 
polar  of  the  focus,  the  ratio  being  equal  to  the  eccen- 
tricity. Hence,  calling  the  focal  distance  />,  and  the 
distance  from  the  polar  (or  directrix)  o,  we  may  write, 
as  the  equation  to  the  Conic, 

p  =  e.d: 

which  will  denote  an  ellipse,  a  parabola,  or  an  hyperbola, 
according  as  e  is  less  than,  equal  to,  or  greater  than  unity. 

G36.  It  follows  from  the  property  mentioned  above, 
that  the  Conic  may  be  defined  as  the  locus  of  a  point 
whose  distance  from  a  fixed  point  is  in  a  constant  ratio 
to  its  distance  from  a  fixed  right  line.  In  fact,  this 
definition  has  been  made  the  basis  of  several  treatises 
upon  the  Conies. 

Calling  the  fixed  point  xfyf9  the  fixed  right  line 
Ax-\-  By  -f-  (7=0,  and  the  variable  point  of  the  curve 
xy,  the  equation  to  the  Conic  is 


If  we  suppose  the  arbitrary  axes  of  this  equation  to  be 
changed  so  that  the  given  line  shall  become  the  axis  of  y, 
and  a  perpendicular  to  it  through  the  given  point  the 


LINEAR  EQUATION  TO  THE  CONIC.  470 

axis  of  x,  we  shall  have  B  =  C  =  0,  y'  =  0,  and  a  new 
value  of  x'  which  may  be  called  2p  :  e.     We  then  get 


This  represents  an  ellipse  when  e  <  1,  a  parabola  when 
e  =  1,  an  hyperbola  when  e  >  1  ;  and,  as  it  can  be 
written 


is  evidently  the  generic  relation  of  which  the  equation 
in  Art.  181  is  a  particular  case. 

Remark.  —  The  formulas  of  this  article  are  only  modi- 
fied expressions  of  the  relation  p  =  e.d',  and  it  is  obvious, 
on  comparing  this  with  (1)  above,  that  the  focal  distance 
of  any  point  on  a  conic  can  always  be  expressed  as  a 
rational  function  of  the  co-ordinates  of  the  point,  in  the 
first  degree. 

We  leave  the  student  to  prove  the  converse  theorem, 
that  a  curve  must  be  a  conic,  if  the  distance  of  every  point 
on  it  from  a  fixed  point  can  be  expressed  as  such  a  rational 
linear  function. 

637.  Linear  Equation  to  the  Conic.  —  It  is  evident 
from  what  has  just  been  said,  that  this  title  would  cor- 
rectly describe  either  the  expression  of  Art.  635  or  the 
modified  form  of  it  given  in  (1)  of  Art.  636.  But  the 
phrase  is  in  fact  reserved  to  designate  a  still  further 
modification  of  the  same  expression,  which  we  will  now 
obtain. 

Suppose  the  origin  of  abscissas  to  be  at  the  focus,  and 
the  axis  of  x  to  be  the  perpendicular  drawn  through  the 
focus  to  its  polar:  the  distance  from  this  polar  (or  direc- 
trix) to  the  point  on  the  curve,  will  then  be  equal  to  the 


480  ANALYTIC  GEOMETRY. 

distance  between  the  directrix  and  the  focus,  increased 
by  the  abscissa  of  the  point;  or,  we  shall  have 


and  the  equation  ft  =  e.d  will  become 
p  =  2p  -f  ex. 

Remark.  —  The   so-called  Linear  Equations  to  the  Ellipse,  the 
Parabola,  and  the  Hyperbola,  namely  (Arts.  360,  457,  568), 


will  all  assume  the  form  just  found,  if  we  shift  their  respective 
origins  to  the  focus,  by  putting  x  —  ae  for  x  in  the  first,  x  +  ae  for 
x  in  the  second,  and  x-\-p  for  x  in  the  third.  We  leave  the  actual 
transformation  to  the  student,  only  reminding  him,  that,  in  the  first 
two  curves,  ±  a  (  I  —  e2)  -—  2p  ;  and  that,  in  the  third,  e  =  l. 

638.  A  form  of  the  preceding  equation  with  which 
the  reader  may  sometimes  meet,  is 

r  =  mx  -f-  n, 

and  any  equation  of  this  form,  in  which  m  and  n  are  any 
two  constants  whatever,  will  denote  a  conic,  whose  eccen- 
tricity will  =  m,  while  its  semi-latus  rectum  will  =  n. 

O39.    Equation  referred  to  Two  Tangents.  —  A  useful  ex- 
pression for  the  Conic  may  be  developed  as  follows  : 

Let  the  equation  to  the  curve,  referred  to  any  axes  whatever,  be 

Ax1  +  2Hxy  +  By*  +  2Gx  +  2Fy  +  C  =  Q  (1). 

To  determine  the  intercepts  of  the  curve  on  the  axes,  we  get,  by 
making  y  and  x  successively  =  0, 

Ax*  +  2Gx  +  C=Q,      Bif  +  2Fy  +  C=  0. 

But  if  the  axes  are  tangents,  the  two  intercepts  on  each  will  be 
equal,  these  quadratics  will  have  equal  roots,  and  we  shall  have 


CONIC  EEFEREED  TO  T  \VO  TANGENTS.        481 

Putting  into   (1)  the  values  of  A  and  B  which  these  conditions 
ive, 

ZCHxy  +  FY  +  2GCx  -f  2FCy  +  C2  =  0. 


Whence,  by  adding  2FGxy  —  2FGxy,  and  re-arranging  the  terms, 
(Gx  +  *V  +  C)*  =  2  (FG  -  C//)  ary. 

This  is  the  equation  we  are  seeking;  and,  as  the  co-efficient  of  xy 
in  it  is  arbitrary  with  respect  to  Gf,  F,  C]  we  may  write  it 

(Gx  +  Fy  +  C?  =  Mxy  (2), 

where  G,  F,  C,  M  are  any  four  constants  whatever. 

Corollary  1.  —  Making  y  and  x  successively  =  0  in  this  equation, 
we  get  the  distances  of  the  two  points  of  contact  from  the  origin, 
namely, 

C  C 

~G>    y~     ~F' 

Calling  the  first  of  these  distances  a,  and  the  second  K,  we  have 
<?  =  -£,     ^=_£; 

a  '  K. 

and  may  write  the  equation  in  the  more  convenient  form 

<3>- 

Corollary  2.  —  The  special  modification  of  this  which  represents 
a  parabola,  deserves  a  separate  notice.  In  order  that  (3)  may 
denote  a  parabola,  we  must  have  (Art.  191) 


F2__     T     JL. 
La/c       ^J        aV  : 


o.  condition  satisfied  by  either  p.  =  0  or  /z  =  4  :  a/c.     If  //  =  0,  the 
equation  becomes 


and  denotes  the  chord  of  contact  of  the  tangent  axes.     If  ft  —  4  :  a/c, 
we  get,  by  taking  the  square  root  of  both  members  of  (3), 


482  ANALYTIC  GEOMETRY. 

or,  after  transposing  and  again  taking  the  square  root, 


an  equation  which  is  sometimes  written 

T//O;  -|-  Vay  =  1/a/c. 

G4O.  Polar  Equation  to  the  Conic.  —  By  compar- 
ing Arts.  443,  535,  627,  it  becomes  evident  that  the 
Conic  may  be  represented  by  the  general  equation 

2^  QN 

1  —  e  cos  6 

In  this,  as  the  reader  will  see  by  referring  to  the  original 
investigations  (Arts.  152,  172,  183),  the  pole  is  at  the 
focus,  and  the  vectorial  angle  6  is  reckoned  from  the 
remote  vertex. 

A  more  useful  expression,  however,  and  the  only  one 
universally  applicable  in  Astronomy,  is 

P~~=I-}-ecosd 

Here,  6  is  reckoned  from  the  vertex  nearest  the  focus 
selected  for  the  pole,  and  I  denotes  the  semi-latus  rectum. 
The  conic  represented  is  an  ellipse,  a  parabola,  or  an 
hyperbola,  according  as  e  <  1,  e  =  1,  or  e  >  1. 

641.  The  Conic  as  the  tocus  of  the  Second  Order 
in  General.  —  All  the  Cartesian  equations  that  precede, 
are  only  reduced  forms  of  the  general  and  unconditioned 
equation 

Ax2  +  ZHxy  +  Bf  -f  2Gx  -f  2Fy  +  C=  0, 

which  may  be  converted  into  any  one  of  them  by  a 
proper  transformation  of  co-ordinates,  and  to  whose 
type  they  all  conform. 


SYSTEM  OF  THE  CONICS.  483 

THE  CONICS  IN  SYSTEM,  AS  SUCCESSIVE  PHASES  OF 
ONE  FORMAL  LAW. 

642*  That  the  Conies  are  successive  phases  of  some 
uniform  law,  we  have  already  seen  in  Section  VII  of 
Part  I.  Of  the  nature  of  that  law,  however,  we  were 
there  unable  to  give  any  better  account  than  this  :  that 
it  expressed  itself  in  the  unconditioned  equation  of  the 
second  degree,  and  became  visible  in  a  threefold  series 
of  curves,  determined  by  the  successive  appearance,  in 
that  equation,  of  the  three  conditions 


But  we  have  now  reached  a  position  which  will  enable 
us  to  state  the  law  in  geometric  language,  to  exhibit  the 
elements  of  form  which  it  embodies,  and  to  trace  the 
steps  by  which  those  elements  cause  the  three  curves  to 
appear  in  an  unbroken  series.  And  it  deserves  especial 
mention,  that  this  geometric  statement  of  the  law  is  fur- 
nished by  the  polar  relation,  as  expressed  in  the  definition 
of  Art.  636. 

G43.  The  generic  law  of  form  which  is  designated  by 
the  name  of  The  Conic,  may  therefore  be  stated  as  follows  : 
The  distance  of  a  variable  point  from  a  fixed  point  shall 
be  in  a  constant  ratio  to  its  distance  from  a  given  right  line. 

Expressing  this  law  in  the  equation  (Art.  637) 

P  =  2p  +  ex  (1), 

let  us  observe  the  development  of  the  system  of  the  three 
curves,  member  after  member,  as  the  given  line  advances 
nearer  and  nearer  to  the  fixed  point. 

We  have  (Art.  637),  for  the  distance  of  the  given  line 
from  the  fixed  point, 

d  =  *JL  (2). 


484  ANALYTIC  GEOMETRY. 

Also,  supposing  a  perpendicular  to  the  given  line  to  be 
drawn  through  the  fixed  point,  the  perpendicular  to  be 
called  the  axis,  and  the  point  in  which  it  cuts  the  curve 
to  be  called  the  vertex,  we  get,  for  the  distance  of  this 
vertex  from  the  fixed  point,  by  making  x  =  —  p  in  equa- 
tion (1), 


Let  the  generation  of  the  system  begin  with  the  given 
line  at  an  infinite  distance  from  the  fixed  point.  In  that 
case,  from  (2),  we  shall  have  e  =  Q.  Under  this  suppo- 
sition, equation  (1)  becomes 


which  (Art.  138,  Cor.  2)  denotes  a  circle,  described  from 
the  fixed  point  as  a  center,  with  a  radius  =  the  semi-latus 
rectum  of  the  Conic  :  a  result  confirmed  by  the  fact,  that, 
under  the  same  supposition,  equation  (3)  becomes  p'=2p-, 
or,  the  distance  of  the  vertex  from  the  focus  becomes  equal 
to  the  radius. 

Now  let  the  given  line  move  parallel  to  itself  along  the 
axis,  assuming  successive  finite  distances  from  the  fixed 
point,  but  with  the  condition  that  every  distance  shall  be 
greater  than  2p.  Then,  from  (2),  e  <  1  ;  and,  from  (3), 
p'  <  2p  and  >  p  :  so  that  a  continuous  series  of  ellipses 
will  appear,  of  ever-increasing  eccentricity,  but  with  a 
constant  latus  rectum,  their  vertices  all  lying  within  a 
segment  of  the  axis  —  p,  contained  between  the  points 
reached  by  measuring  from  the  fixed  point  distances  =p 
and  2p. 

Next,  let  the  given  line  have  attained  the  distance  =  2jt? 
from  the  fixed  point.  We  shall  then  have,  from  (2),  e  =  1  ; 
and,  from  (3),  p'  —p.  That  is,  we  shall  have  &  parabola, 
described  upon  the  constant  latus  rectum. 


GENERATION  OF  THE  SYSTEM.  485 

Finally,  let  the  given  line  advance  from  its  last  posi- 
tion, and  approach  the  fixed  point  indefinitely.  Then,  d 
being  less  than  2p,  we  shall  have,  from  (2),  e  >>  1 ;  and, 
from  (3),  f/<p:  so  that  there  will  arise  a  continuous 
series  of  hyperbolas,  with  a  constant  latus  rectum,  but 
with  an  ever-increasing  eccentricity;  with  their  vertices 
all  lying  within  a  segment  of  the  axis  =  p,  measured  from 
the  fixed  point,  and  with  their  branches  tending  to  coin- 
cide with  the  given  line  as  that  line  tends  toward  the  fixed 
point.  When  the  given  line  attains  the  particular  distance 
=  p  1/2  from  the  fixed  point,  we  shall  have  e  — 1/2;  or, 
the  hyperbola  (Art.  456,  Cor.)  will  be  rectangular. 

Thus,  the  order  of  the  curves,  as  foreshadowed  by 
their  analytic  criteria,  is  verified  by  a  systematic  gen- 
eration. 

644.  The  results  of  the  preceding  article  may  be 
tabulated  as  follows: 

{e  =  0  .'.  Circle. 
e<l.:  Eccentric. 

TO      0     Semi-latus  rectum  =  Distance  of  Focal  Polar  }e==i 
.-.  PARABOLA.  J 

(  e  >1  .-.  Oblique. 
.  Semi-latus  rectum  >  Distance  of  Focal  Polar  J 

.-.HYPERBOLA.  U^vT.-.  Rect'r. 

PROPERTIES  OF  THE  CONIC  IN  GENERAL. 

645.  The  views  thus  far  taken  of  the  Conic  in  the 
present   Chapter,    although   generic,   have    nevertheless 
been   obtained  from   a  standpoint  not  strictly  analytic. 
For   our  results  have  been  derived  from  a  comparison 
of  the  properties  in  which  the  three  curves,  after  sepa- 
rate treatment  by  means  of  equations  based  upon  certain 
assumed  properties,  have  been  found  to  agree.     But  we 
shall  now,  for  a  few  pages,  ascend  to  the  strictly  analytic 


486  ANALYTIC  GEOMETRY. 

point   of  view,   and,   beginning   with  the  unconditioned 
equation 

Ax2  +  ZHxy  +  Bf  +  2Gx  -f  ZFy  +  (7=0, 


shall  show  how  the  properties  common  to  the  System 
of  the  Conies,  or  peculiar  to  its  several  members,  may 
be  developed  from  this  abstract  symbol,  without  assum- 
ing a  single  one  of  them. 


THE  POLAR  RELATION. 

646.  Intersection  of  the  Conic  with  the  Right 

JLiiie.  —  If  we  eliminate  between 


and  y  =  mx  -f-  &,  we  shall  obviously  get  a  quadratic  in 
x  to  determine  the  abscissa  of  the  point  in  which  the 
Conic  cuts  any  right  line.  Hence,  Every  right  line 
meets  the  Conic  in  two  points,  real,  coincident,  or  imag- 
inary. 

In  particular,  for  the  points  in  which  the  curve  meets 
the  axes  of  reference,  we  get,  by  making  y  and  x  in  (1) 
successively  =  0,  the  determining  quadratics 

Ax2  -f  2Gx  -\-  C=  0,      By2  -f  2Fy  -j-  C=  0      (2). 

647.  The  Chord  of  the  Conic.— If  a  right  line 
meets  the  Conic  in  two  real  points,  x'y'  and  xrfyn,  we 
may  write  its  equation 

A  (x—xf)  (x—xir)  +  2H(x—x')  (y—y")  -f  B  (y—yr)  (y—y") 


For  this  is  the  equation  to  some  right  line,  since,  upon 
expansion,  its  terms  of  the  second  degree  destroy  each 


TANGENT  OF  THE  CONIC.  487 

other  ;  and  to  a  line  that  passes  through  the  points  x'y'  , 
x"y"  of  the  curve,  because  if  x'  and  y1  or  x"  and  y"  be 
substituted  for  x  and  y  in  it,  we  either  get 


or  else 


which  are  simply  the  conditions  that  x'y',  x"y"  may  be 
on  the  curve. 

648.  The  Tangent  of  the  Conic.  —  Making  x"  =  x', 
and  y"  =  yr,  in  the  preceding  equation  to  the  chord,  we 
get  the  equation  to  the  tangent, 


—  yj 
=  Ax2  +  IHxy  +  Ef  +  2  Gx  +  2Fy  +  C\ 


which,  after  expansion,  assumes  the  form 

ZAx'x  +  2H(x'y  +  y'x)  +  ZBy'y  +  2Gx  +  2Fy  +  C 


Adding  2Gxf  +  %Fij  +  (7  to  both  members  of  this,  and 
remembering  that  the  point  of  contact  x'y1  must  satisfy 
the  equation  to  the  Conic,  we  get  the  usual  form  of  the 
equation  to  the  tangent,  namely, 


*/')  +  ^=0         (1). 

By  expanding,  and  re-collecting  the  terms,  this  may  be 
otherwise  written 


JFy+<7=0         (2). 
An.  Ge.  44. 


ANALYTIC  GEOMETRY. 

These  equations  express  the  law  which  always  con- 
nects the  co-ordinates  of  any  point  on  the  tangent  with 
those  of  the  point  of  contact.  Hence,  if  a  point  through 
which  to  draw  a  tangent  were  given,  and  the  point  of 
contact  were  required,  equation  (1)  would  still  express 
the  relation  between  the  co-ordinates  of  these  points, 
only  x'y'  would  then  denote  the  given  point  through 
which  the  tangent  would  pass,  and  xy  the  required  point 
of  contact.  That  is,  the  equation  which  when  x'y'  is  on 
the  curve  represents  the  tangent  at  x'y',  when  x'y'  is  sit- 
uated elsewhere  denotes  a  right  line  on  which  will  be 
found  the  point  of  contact  of  the  tangent  drawn  through 
x'y'  .  Now  this  line,  in  common  with  every  other,  meets 
the  Conic  in  two  points:  hence,  From  any  given  point, 
there.  can  be  drawn  to  the  Conic  two  tangents,  real,  coin- 
cident, or  imaginary. 

We  thus  learn  that  our  curve  is  of  the  Second  class  as 
well  as  of  the  Second  order. 

649.  Chord  of  Contact  in  the  Conic.  —  From  what 
has  just  been  stated,  it  follows  that 


or  its  equivalent  form  (2)  above,  is  the  equation  to  the 
chord  of  contact  of  the  two  tangents  drawn  through  x'y'. 

65O.  -ILociis  of  the  Intersection  of  Tangents  whose 
Chord  of  Contact  revolves  about  a  Fixed  Point.  —  Let 

x'yf  be  the  fixed  point,  and  #,?/,  the  intersection  of  the 
two  tangents  corresponding  to  the  chord.  Then,  as  x'y1 
is  by  supposition  always  on  the  revolving  chord  of  con- 
tact, we  shall  have  the  condition 


H(xiyf 

i  +  y')  +  0=  0, 


POLAR  IN  THE  CONIC.  489 

irrespective  of  the  direction  of  the  chord.  In  other 
words,  the  intersection  of  the  tangents  will  always  be 
found  upon  a  right  line  whose  equation  is 

Ax'x  +  H(xfy  -f  y'x)  +  By'y 

+  G(x  +  <x/)  +  F(y  +  y>)  +  0=  0. 

651.  The  Point  and  the  Right  Une,  Reciprocals 
with  respect  to  the  Conic.  —  -The  result  of  the  preceding 
article  may  be  stated  as  follows  :  If  through  a  fixed  point 
chords  be  drawn  to  the  Conic,  and  tangents  be  formed  at 
the  extremities  of  each  chord,  the  intersections  of  the  sev- 
eral pairs  of  tangents  will  lie  on  one  right  line. 

Also,  we  may  write  the  equation  to  the  chord  of  con- 
tact of  two  tangents  drawn  through  x'y'  (Art.  649) 

(Ax'  +  Hy'  +  G)  x  +  (ffz'  +  By'  +  F)y 


so  that  (Art.  117),  if  we  suppose  x'y'  to  move  along  a 
given  right  line,  the  chord  of  contact  will  revolve  about 
a  fixed  point.  In  other  words  :  If  from  different  points 
lying  on  one  right  line  pairs  of  tangents  be  drawn  to  the 
Conic,  their  several  chords  of  contact  will  meet  in  one 
point. 

Combining  these  two  properties,  we  see  that  our  curve 
imparts  to  every  point  in  its  plane,  the  power  of  deter- 
mining a  right  line  ;  and  to  every  right  line,  the  power 
of  determining  a  point.  That  is,  it  renders  the  Point 
and  the  Right  Line  reciprocal  forms. 

652.  The  Polar  ami  its  Equation.  —  We  perceive, 
then,  that  the  relation  between  a  tangent  and  its  point 
of  contact,  and  the  relation  between  the  chord  of  contact 
and  the  intersection  of  the  corresponding  tangents,  are 
only  particular  cases  of  a  general  law  which,  with  respect 


490  ANALYTIC  GEOMETRY. 

to  the  Conic,  connects  any  fixed  point  with  a  correspond- 
ing right  line.  From  the  result  of  the  last  article,  more- 
over, it  appears  that  we  shall  fitly  express  this  law  by 
calling  the  line  which  corresponds  to  any  point,  the  polar 
(i.  e.  the  reciprocal)  of  the  point,  and  the  point  itself  the 
pole  of  the  line. 

We  are  henceforth,  then,  to  consider  the  equation 

Ax'x  +  H(x'y  +  y'x)  +  By'y 


as  in  general  denoting  the  polar  of  x'y' ;  and  must  regard 
the  tangent  at  x'y'  as  the  position  assumed  by  the  polar 
when  x'y'  is  on  the  Conic. 

653.  The  Conic  referred  to  its  Axis  and  Ver- 
tex.— Before  taking  out  any  additional  properties  of  the 
curve,  it  will  be  best  to  reduce  the  general  equation 

Ax2  +  ZHxy  +  By2  +  2Gx  +  2Fy  +  (7=0 

to  a  simpler  form.  Supposing  the  axes  of  reference  to 
be  rectangular,  let  us  revolve  them  through  an  angle  #, 
such  that 

tan  26  = 


A  —  B 

We  shall  thus  (Art.  156)  destroy  the  co-efficient  of  xy, 
and  the  equation  will  assume  the  form 

Ax2  +  B'y2  +  2G'x  +  ZF'y  +  (7=0. 

If  in  addition  we  remove  the  origin  to  a  point  x'y' ,  we 
shall  get  (Art.  163,  Th.  I) 

Arx2  +  B'y2  +  2  (A'xr  +  G')x  +  2  (B'y'  +  F')  y 

+  (A'x12  +  B'y'2  -}-  ZG'x'  +  ZF'y'  +  C)  =  0. 


VERTICAL  EQUATION  TO  THE  CONIC.          491 

In  order,  then,  that  the  constant  term  and  the  co-efficient 
of  y  may  vanish  together,  we  must  have  simultaneously 

A'x'2  +  B'y'2  +  2  G'x'  +  ZF'y'  +  (7=0,      B'y'  +  F  =  0  ; 


that  is,  we  must  take  the  new  origin  #y  at  the  inter- 
section of  the  curve  with  the  right  line  By-\-F=Q. 
Making  this  change  of  origin,  our  equation  becomes 


which,  by  putting  20"  :  B'  =-  —  P,  and  A'  :  B'  =  -  R, 
and  transposing,  may  be  written 


Here,  for  every  value  of  #,  there  will  be  two  values 
of  y,  numerically  equal  with  opposite  signs  :  the  curve 
is  therefore  symmetric  to  the  new  axis  of  x,  which  for 
that  reason  shall  be  called  an  axis  of  the  curve.  If  we 
seek  the  intersections  of  the  curve  with  the  new  axis  of 
y,  by  making  x  =  0  in  the  equation,  we  get  y  —  ±  0  ;  so 
that  the  new  axis  of  y  meets  the  curve  in  two  coincident 
points  at  the  origin  ;  or,  in  other  words,  is  tangent  to  the 
curve  at  the  origin.  Besides,  the  new  axes  are  rectan- 
gular: hence,  combining  this  fact  with  those  just  estab- 
lished, the  new  origin  is  the  extreme  point,  or  vertex,  of 
the  curve  ;  and  we  learn  that  our  axes  of  reference  are 
the  principal  axis  of  the  curve  and  the  tangent  at  its 
vertex.  And,  in  fact,  our  new  equation  is  identical  with 
that  obtained  in  Art.  634. 

654.  Focus  of  the  Conic,  and  its  Polar.  —  Taking 
up  our  equation  in  its  new  form 

(1), 


492  ANALYTIC  GEOMETRY. 

let  e  be  such  a  quantity  that 

e2  =  I  +  R  (2), 

and  let  that  point  whose  co-ordinates  are 


be  called  the  focus  of  the  Conic. 

The  equation  to  the  polar  of  any  point,  referred  to  our 
present  axes,  is  at  once  found  from  the  general  equation 
of  Art.  652,  by  putting  for  A,  H,  B,  G,  F,  C  their  values 
as  given  by  (1).  It  is  therefore 

P(x  +  x')  (4). 


Hence,  substituting  for  xf  and  y'  the  values  given  in 
(3),  we  get,  for  the  polar  of  the  focus, 

or,  after  replacing  R  by  its  value  e2  —  1  from  (2), 

x=     -         P  (6) 

Equation  (6)  shows  that  the  polar  of  the  focus  is  per- 
*  pendicular  to  the  axis  of  the  Conic,  and  cuts  it  on  the 
opposite  side  of  the  vertex  from  the  focus,  at  a  distance 
—  an  eih  part  of  the  distance  of  the  focus.  And  we 
shall  see,,  in  a  moment,  that  the  ratio  thus  found  between 
the  distances  of  the  vertex  from  the  focus  and  from  its 
polar,  subsists  between  the  distances  of  any  point  on  the 
curve  from  those  two  limits. 

From  (3)  we  have  (Art.  51, 1,  Cor.  1),  for  the  distance  p 
of  any  point  xy  from  the  focus, 


THE  FOCUS  AND  ITS  POLAR.  493 

since  xy  is  on  the  Conic.    Replacing  R  by  its  value  e2  —  1, 
and  reducing,  we  get 


•  2(1  +  ,) 

Also,  for  the  distance  from  xy  to  the  polar  of  the  focus, 
we  have,  from  (6)  by  Art.  105,  Cor.  2, 

,       2  (1  +  e)  ex  +  P 
2(1+,), 

Hence,  dividing  (7)  by  (8),  we  get 

H  (9). 

That  is,  The  distance  of  any  point  on  the  Conic  from  the 
focus,  is  in  a  constant  ratio  to  its  distance  from  the  polar 
of  the  focus. 

The  ratio  e,  we  will  call  the  eccentricity  of  the  Conic. 

655.  The  Species  of  the  Conic,  and  their  Fig- 
ures. —  The  preceding  investigation  leads  directly  to  the 
resolution  of  the  vague  and  general  Conic  into  three 
specific  curves,  and  the  generic  property  just  developed 
will  enable  us  at  once  to  determine  the  figures  of  these. 

For  since  e2  =  1  -(-  R,  we  shall  evidently  have 


according  as  R  is  negative,  equal  to  zero,  or  positive. 
Therefore,  by  embodying  the  property  of  (9)  in  the 
mechanical  contrivances  described  in  the  corollaries  to 
Arts.  439,  532,  618,  we  can  generate  three  distinct 
curves,  depending  on  the  value  of  e  ;  as  follows  : 

In  the  first,  e  will  fall  short  of  1  :  whence  the  curve 
may  be  called  an  ellipse. 


494  ANAL  YTIC  GEOMETR  Y. 

In  the  second,  e  will  equal  1  :  whence  the  curve  may 
be  called  a  parabola. 

In  the  third,  e  will  exceed  1  :  whence  the  curve  may 
be  called  an  hyperbola. 

The  figures  of  these  curves  are  therefore  such  as  the 
methods  of  generation  give,  and  need  not  be  drawn  here, 
as  they  are  already  familiar. 

If  we  put  x=P  :  2  (1  -f  e)  in  equation  (1)  of  the  pre- 
ceding article,  we  get,  for  the  ordinate  erected  at  the 
focus, 

*  =  --: 

y      -  2 

whence,  calling  the  double  ordinate  through  the  focus 
the  latus  rectum, 

latus  rectum  =  P  (1). 

We  thus  obtain  a  significant  interpretation  for  the  par- 
ameter P  of  our  equation  ;  but  we  do  more.  For,  by 
the  generic  property  of  the  preceding  article,  the  distance 
of  the  focus  from  its  polar  must  equal  an  eth  part  of  its 
distance  from  the  extremity  of  the  latus  rectum,  and 
therefore  can  now  be  expressed  by 


Hence,  when  e  <  1,  the  semi-latus  rectum  will  be  less 
than  the  distance  of  the  focus  from  its  polar  ;  when 
e  =  1,  it  will  be  equal  to  that  distance  ;  and  when  e  >  1, 
it  will  be  greater  than  that  distance.  In  other  words, 
the  classification  reached  above,  is  identical  with  that 
of  Art.  644  :  as  may  be  further  shown  by  the  fact,  that, 
if  R  =  —  1,  e  =  0  ;  and  if  R  =  +  1,  e  =  1/2. 


DIAMETERS  AND  THE  CENTER.  495 

Also,  when  e  —  0,  equation  (7)  of  the  preceding  article 
gives 


That  is,  when  e  =  0,  the  curve  is  such  that  all  its  points 
are  equally  distant  from  the  focus,  or  its  figure  is  that 
of  the  Circle.  Hence,  as  e  increases  from  0  toward  oo, 
the  figure  of  the  curve  may  be  supposed  to  deviate  more 
and  more  from  the  circular  form,  and  we  see  the  pro- 
priety of  calling  e  the  eccentricity. 

DIAMETERS  AND  THE  CENTER. 

656.  A  very  significant  question  in  regard  to  any 
curve  is,  What  is  the  form  of  its  diameters,  that  is,  of  the 
lines  that  bisect  systems  of  parallel  chords  in  it  ?    Let  us, 
then,  settle  this  question  for  the  Conic. 

657.  Equation  to  any  Diameter.  —  If  we  suppose  6' 
to  be  the  common  inclination  of  any  system  of  parallel 
chords,  x'y'  the  intersection  of  any  member  of  the  system 
with  the  Conic,  and  xy  its  middle  point,  we  shall  have 
(Art.  102) 

xf  =  x  —  Zcos#',      y'—y 


where  I  is  the  distance  from  xy  to  x'y'.     But  since  x'y' 
is  on  the  Conic,  we  get,  by  equation  (1)  of  Art.  646, 


A(x  — 

C=0. 


Expanding,  collecting  terms,  and  putting  S  for  the  first 
member  of  the  general  equation  of  the  second  degree, 
we  get 

(A  cos2  0"  +  2Hcos  tf  sin  0'  +  B  sin2  P)  I'2 
—  2[(Ax  +  H 

An.  Ge.  45. 


496  ANALYTIC  GEOMETRY. 

Now  xij  being  the  middle  point  of  a  chord,  the  two 
values  of  I  given  by  this  quadratic  must  be  numerically 
equal  with  opposite  signs.  Hence,  (Alg.,  234,  Prop.  3d,) 
the  co-efficient  of  I  vanishes,  and  we  obtain,  as  the  equa- 
tion to  any  diameter, 

(Ax  -f-  Hy  +  G)  +  (Hx  +  By  -f  F)  tan  '  0'  =  0, 

in  which  6f  is  the  inclination  of  the  chords  which  the 
diameter  bisects. 

658.  Form  and  Position  of  IMamcters.  —  Com- 
paring the  equation  just  obtained  with  that  of  Art.  108, 
we  learn  that  every  diameter  of  the  Conic  is  a  right 
line,  and  passes  through  the  intersection  of  the  two 
lines 

G  =  Q,     Hx  +  By  +  F  =  0  ; 


that  is,  through  the  point  whose  co-ordinates  (Art.  106) 
are 

BG—HF  _  AF—HG        m 

~  H*  -  AB  '          ~  H* 


Moreover,  putting  0  =  the  inclination  of  any  diameter, 
we  have  (Art.  108) 

A  -f  H  tan  0' 


so  that,  as  6'  is  arbitrary,  a  diameter  may  have  any  in- 
clination whatever  to  the  axis  of  x;  or,  every  right  line 
that  passes  through  the  point  (1)  is  a  diameter.  Hence, 
as  (1)  is  in  turn  upon  every  diameter,  it  is  the  middle 
point  of  every  chord  drawn  through  it,  and  may  there- 
fore be  called  the  center  of  the  Conic. 

For  the  form  and  position  of  conic  diameters  in  gen- 
eral, we  therefore  have  the  two  theorems  :  .Every  diam- 
eter is  a  right  line  passing  through  the  Center;  and,  Every 
right  line  that  passes  through  the  center  is  a  diameter. 


DIAMETERS  AND  THE  CENTER.  497 


Hence,  the  lines  Ax  +  ffi/+G  = 
are  both  diameters  ;  and,  by  making  6r  —  0  in  the  final 
equation  of  Art.  657,  we  learn  that  the  former  bisects 
chords  parallel  to  the  axis  of  x  ;  while,  by  making  0'=  90°, 
we  see  that  the  latter  bisects  chords  parallel  to  the  axis 
of?/. 

From  (1)  we  see  that  the  center  of  the  Conic  will  be 
at  a  finite  distance  from  the  origin,  so  long  as  H2  —  AB 
is  not  equal  to  zero  ;  but  will  recede  to  infinity,  if  H2=AB. 
Now,  by  putting  for  A,  B,  F,  G,  H  the  values  they  have 
.when  the  equation  to  the  Conic  takes  the  form 

y2  =  Px  +  Rx\ 

we  get  the  co-ordinates  of  the  center,  referred  to  the  prin- 
cipal axis  and  its  vertical  tangent,  namely, 

*  =  -4     ,  =  0  (3), 

which  show  that  the  center  is  situated  on  the  principal 
axis,  at  a  distance  from  the  vertex  =  —  Pi  2R.  This 
distance,  then,  will  be  finite  if  R  is  either  positive  or 
negative,  but  infinite  if  R  =  0.  Hence,  (Art.  655,)  the 
diameters  of  the  two  curves  which  we  have  named  the 
Ellipse  and  the  Hyperbola,  meet  in  a  finite  point,  and 
are  inclined  to  each  other;  but  the  diameters  of  the 
curve  called  the  Parabola  meet  only  at  infinity,  or,  in 
other  words,  are  all  parallel:  a  result  corroborated  by 
the  fact,  that,  if  in  (2)  we  replace  A  by  the  value  H2  :  B, 
which  it  will  have  if  H2  —  AB,  we  get 

tan0=    -J  (4), 

showing  that  all  the  diameters  of  any  given  parabola  arc 
equally  inclined  to  the  axis  of  x. 


498  ANALYTIC  GEOMETRY. 

659.  Farther  Classificati©n  of  the  Conic.— It  thus 
appears  that  the  Ellipse  and  the  Hyperbola  may  be 
classed  together  as  central  conies,  while  the  Parabola 
may  be  styled  the  non-central  conic.  Adding  this  prior 
subdivision,  the  table  of  Art.  644  will  appear  thus : 


Semi-latus  Rectum  <  Dist.  of  Foe.       "    ' 


.-.  ELLIPSE.  (  e<l  .-.  Eccent. 

CENTRAL  • 

Semi-latus  Rectum  >  Dist.  of  Foe.  Pol.   (  e  >  l  •'•  Oblique. 
.-.  HYPERBOLA.  |  e  =  1/2".-.  Rect'r. 


NON-      f  Semi-latus  Rectum  =  Dist.  of  Foe.  Pol. 
CENTRAL  1  .'.  PARABOLA. 


CONJUGATE  DIAMETERS  AND  THE  AXES. 

66O.  Relative  inclination  of  Diameters  and  their 
Ordinates. — The  halves  of  the  chords  which  a  diameter 
bisects  may  be  called  its  ordinates.  If,  then,  6  =  the 
inclination  of  any  diameter,  and  6'  =  that  of  its  ordi- 
nates, we  have,  from  (2)  of  Art.  658, 

£tan  0  tan  6'  +  JT(tan  0  -f  tan  d')  +  A  =  Q 

as  the  relation  always  connecting  the  inclinations  of  a 
diameter  and  its  ordinates. 

Now  this  may  either  be  read  as  the  condition  that  the 
diameter  having  the  inclination  6  may  bisect  chords 
having  the  inclination  61 ',  or  vice  versa.  Hence,.  If  a 
diameter  bisect  chords  parallel  to  a  second,  the  second 
will  bisect  chords  parallel  to  the  first. 

This  property  is  however  restricted  to  the  central 
conies ;  for  it  is  impossible  that  the  ordinates  of  any 
parabolic  diameter  should  be  parallel  to  a  second,  since 
all  parabolic  diameters  are  parallel  to  each  other. 


AXES  OF  THE  CONIC.  499 

661.  Condition    that    two   Diameters   be  Conju- 
gate.— We  indicate  that  two  diameters  of  a  central  conic 
are   in   the  relation  above-mentioned,  by   calling  them 
conjugate  diameters.     Since,  then,  the  conjugate  of  any 
diameter  is  parallel  to  its  ordinates,  by  interpreting  6 
and  6f  as  the  inclinations  of  two  diameters, 

B  tan  6  tan  Of  +  IT  (tan  6  +  tan  6')  +  A  =  0 

becomes  the  condition  that  the  two  diameters  may  be 
conjugate. 

662.  The  Axes,  and  their  liquation. — The  condi- 
tion just   established  may  be  referred  to  the  principal 
axis  and  vertex  of  the  Conic  by  putting  for  A,B,  ^T  the 
values  they  have   in  the  equation  y2  =  Px  +  fix2.     It 
thus  becomes 

tan  0  tan/?'  =  E  (1). 

In  this,  if  we  suppose  6  =  0,  but  not  otherwise,  we 

get 

tan  6f  =  GO, 

and  learn  that  the  conjugate  of  the  principal  axis  is 
perpendicular  to  it.  Hence.  In  a  central  conic  there  is 
one,  and  but  one,  pair  of  rectangular  conjugates.* 

We  will  call  these  rectangular  conjugates  the  axes  of 
the  conic.  The  one  hitherto  named  the  principal,  shall 
now  be  termed  the  transverse  axis ;  and  the  other,  the 
conjugate  axis.  Their  respective  equations,  referred  to 
the  same  system  as  the  conic  y2  =  Px  -f-  Jlx2,  will  be 

P=Q  (2). 


#  Unless  the  conic  is  a  circle  :  when  R  will  =  —  1,  and  the  condition 
(1)  will  become  1  +  tan  9  tan  0'  =  0  ;  so  that  (Art.  96,  Cor.  1)  all  the  con- 
jugates will  be  at  right  angles. 


500  ANALYTIC  GEOMETRY. 

For  the  first  equation  represents  the  axis  of  x  ;  and  the 
second,  a  perpendicular  to  it  passing  through  the  center. 

Hence, 

P)y  =  0  (3) 


is  the   equation   to   both   axes,  in  the  same  system  of 
reference. 

663.  Equation    to    the    Conic,    in    its    Simplest 
Forms.  —  If  in  the  equation 


(1), 

we  suppose  JR=Q  .  •  .  e  =  l,  the  quantity  P  :  2  (1  -f  e), 
which  by  (3)  of  Art.  654  denotes  the  distance  of  the 
focus  from  the  vertex,  will  become  =  ^  P:  showing  that 
in  the  Parabola  the  latus  rectum  P  is  equal  to  four  times 
the  focal  distance  of  the  vertex.  Putting  this  latter  dis- 
tance —  JP,  and  giving  to  R  in  (1)  its  corresponding 
value  —  0,  the  equation  to  the  Parabola  will  be 

(2). 


But  if  R  be  positive  or  negative,  or  (1)  denote  the 
Ellipse  or  the  Hyperbola,  this  simplification  is  impossi- 
ble. If,  however,  we  transform  (1)  to  the  center  and 
axes,  by  putting  [x  —  (P  :  27£)]  for  x9  we  get 


or,  after  obvious  reductions, 

4E2x2  —  42  =  P2. 


Here,  making  y  and  x  successively  =  0,  we  obtain,  for 
the  lengths  of  the  semi-axes, 

p  P  r~r 

X  =  -r-  _  ,  77  —  -f-       \  I  _ 

"212'       y       ~2V      R- 


ASYMPTOTES  OF  THE  CONIC.  501 

Putting  a  to  denote  the  first  of  these  lengths,  and  b  to 
denote  the  second,  we  get 


:,       «Z  =  -£  (A), 


and  the  central  equation  becomes 


the  upper  sign  corresponding  to  —  R,  and  the  lower  to 
+  R. 

Equations  (2)  and  (3)  are  the  simplest  forms  of  the 
equation  to  the  Conic.  In  them,  we  have  reached  the 
same  forms  with  which  we  set  out  upon  the  separate  in- 
vestigation of  the  Parabola,  the  Ellipse,  and  the  Hyper- 
bola. Of  course,  then,  we  can  now  develop  all  the  prop- 
erties derived  from  them  in  the  preceding  Chapters,  and 
the  reader  will  be  convinced  of  the  adequacy  of  the  purely 
analytic  method  without  proceeding  farther.  We  will 
therefore  present  but  a  single  topic  more,  whose  treat- 
ment from  the  generic  point  of  view  has  an  especial 
interest. 

THE   ASYMPTOTES. 

664.  We  have  shown  (Art.  646)  that  every  right  line 
meets  the  Conic  in  two  points,  real,  coincident,  or  imag- 
inary. A  particular  case  of  the  real  intersections  deserves 
notice. 

The  quadratic  by  which  we  determine  the  intersections 
of  a  right  line  with  the  Conic,  may  sometimes  take  the 
form  of  a  simple  equation,  by  reason  of  the  absence  of 
the  co-efficient  A  or  B  in  the  equation  to  the  Conic. 
Thus 

Hxy    +  Bif  +  2Gx  +  2Fy+C=  0         (1) 


502  ANALYTIC  GEOMETRY. 

gives,  on  making  y  =  0,  only  the  simple  equation- 

2Gx+C=Q  (2) 

to  determine  the  intersections  of  the  curve  with  the  axis 
of  x,  apparently  indicating  but  a  single  intersection.  In 
fact,  it  does  indicate  a  single  finite  intersection;  but  it 
is  a  settled  principle  of  analysis  (Alg.,  238)  that  an 
equation  arising  in  the  manner  (2)  does,  shall  be  re- 
garded as  a  quadratic  of  the  form 

Ox2  -f  2Hx  +  5  =  0, 

one  of  whose  roots  is  finite,  and  the  other  infinite. 
Hence,  the  consistent  interpretation  of  such  an  equation 
as  (2),  will  be  that  the  corresponding  line  meets  the 
Conic  in  one  finite  point  and  in  one  point  infinitely 
distant  from  the  origin. 

665.  Transforming  the  general    equation    to    polar 
co-ordinates,  we  obtain 

(A  cos2  6+2H  cos  6  sin  0  +  S  sin2  6)  f)2 

+  2(#cos0  +  .Fsin0)/>  +  C=0        (1). 

The  condition,  then,  that  the  radius  vector  may  meet  the 
Conic  at  infinity  is 

A  cos2  0  -f  2JF/cos  6  sin  0  +  B  sin2  0  =  0        (2) : 

a  quadratic  in  6,  and  therefore  satisfied  by  two  values 
of  the  vectorial  angle,  which  evidently  will  be  real,  equal, 
or  imaginary,  according  as  If2  —  AB  is  greater  than, 
equal  to,  or  less  than  zero.  Hence,  as  the  origin  may 
be  taken  at  any  point,  Through  any  given  point  there  can 
be  drawn  two  real,  coincident,  or  imaginary  lines  which 
will  meet  the  Conic  at  infinity. 


ASYMPTOTES  OF  THE  CONIC.  503 

Moreover,  since  a  change  of  origin  (Art.  163,  Th.  I) 
does  not  affect  the  co-efficients  A,  H,  B,  the  directions 
of  these  lines  for  any  given  conic  will  in  all  cases  be 
determined  by  the  same  quadratic  (2).  That  is,  All  lines 
that  meet  the  Conic  at  infinity  are  parallel. 

It  is  to  be  noted,  that  in  general  each  of  the  radii 
vectores  determined  by  (2)  also  meets  the  curve  in  one 
finite  point,  whose  position  is  given  by  the  finite  terms 
of  (1),  namely,  by 

2  (G  cos  0  +  F  sin  0)  p  +  C=  0  (3). 

A  convenient  method  of  finding  the  equation  to  the 
two  lines  which  pass  through  the  origin  and  meet  the 
curve  at  infinity,  will  be  to  multiply  (2)  throughout  by 
ft\  and  then  put  x  for  ft  cos  6,  and  y  for  ft  sin  d.  We 
thus  obtain 

Ax2  +  ZHxy  +  By2  =  0  (4), 

the  equation  of  Art.  127.  The  two  lines,  then,  in  case 
the  conic  is  an  ellipse,  will  be  imaginary;  in  case  it  is 
a  parabola,  they  will  be  coincident;  and  in  case  it  is  an 
hyperbola,  they  will  be  real. 

666.  If  we  now  suppose  the  general  equation  to  be 
transformed  to  the  center,  the  co-efficients  G  and  F 
(Art.  163,  Th.  Ill)  will  vanish.  For  that  origin,  then, 
the  condition  (2)  of  the  preceding  article  will  occur 
simultaneously  with  the  disappearance  of  the  co-efficient 
of  ft  in  (1),  and  the  roots  of  the  latter  equation  will 
therefore  be  simultaneously  infinite  and  equal.  Hence, 
Through  the  center  there  can  be  drawn  tivo  lines,  each 
of  which  will  meet  the  Conic  in  two  coincident  points  at 
infinity. 

These  tangents  at  infinity  may  appropriately  be  called 
asymptotes;  since  the  curve  must  converge  to  them  as  it 


504  ANALYTIC  GEOMETRY. 

recedes  to  infinity,  but  can  not  merge  into  them  except 
at  infinity.  Since,  then,  all  the  lines  that  meet  the  curve 
at  infinity  are  parallel,  equation  (4)  of  the  preceding 
article  will  in  general  denote  a  pair  of  parallels  to  the 
asymptotes,  passing  through  the  origin.  Hence,  sup- 
posing the  center  to  be  the  origin,  we  have,  for  the 
equation  to  the  asymptotes, 


Ax*  +  2Hxy  +  Btf  =  0  (1), 

since  the  co-efficients  A,  H,  B  remain  the  same  for  every 

origin.     This    is    the   same   as   saying,   that,  given    any 

central  equation  to  a  conic,  the  asymptotes  are  found  by 

equating  to  zero  its  terms  of  the  second  degree.     The  form 

of  (1)  shows  that  these  lines  are  real  in  the  Hyperbola, 

coincident  in  the  Parabola,  and  imaginary  in  the  Ellipse. 

If  now,  in  the  condition  of'Art.  661,  we  make  tan  6'  '  = 

—  cot  0,  the  corresponding  conjugates  will  be  at  right 

angles  ;  that  is,  they  will  be  the  axes.     But  then 

#tan2  6  +  (A  —  B)  tan  0  —  H=  0. 
Multiply  this  by  />2,  put  x  for  p  cos  0,  and  y  for  p  sin  0  :  then 
Hx*-(A-B)xy-H>f  =  Q  (2), 

the  equation  to  the  axes,  if  the  center  is  origin. 

Now  (2)  is  the  equation  of  Art.  129,  and  therefore 
denotes  two  right  lines  bisecting  the  angles  between  the 
lines  represented  by  (1).  Hence,  The  axes  bisect  the 
angles  between  the  asymptotes,  and  are  real  whether  the 
asymptotes  are  real  or  imaginary. 

CONDITIONS   DETERMINING  A   CONIC. 

The  general  equation  of  the  second  degree, 
Ax2  -}-  2Hxy  +  By- 


CONDITIONS  DETERMINING  A  CONIC.          505 

may  of  course  be  divided  through  by  any  one  of  its  co- 
efficients, and  therefore  contains  five,  and  only  five, 
arbitrary  constants.  Hence,  Five  conditions  are  neces- 
sary and  sufficient  to  determine  a  conic. 

Thus,  a  conic  may  be  made  to  pass  through  five  given 
points  ;  or,  to  pass  through  four  points  and  touch  a  given 
line  ;  or,  to  pass  through  three  points  and  touch  two  given 
lines  ;  etc.  And  in  case  the  equation  to  a  conic  contains 
less  than  five  constants,  we  must  understand  that  the 
curve  has  already  been  subjected  to  a  series  of  condi- 
tions, equal  in  number  to  the  difference  between  five 
and  the  number  of  constants  in  its  equation.  Thus,  the 
conic 

y*  =  Px  +  llx2 

has  already  been  subjected  to  three  conditions;  namely, 
passing  through  a  given  point  (the  vertex),  touching  a 
given  line  (the  axis  of  ?/),  and  having  the  focus  on  a 
given  line  (the  axis  of  x). 


The  solution  of  two  general  problems  which  are 
often  of  use  in  connection  with  conies,  may  conveniently 
be  presented  here. 

I.  To  determine  the  relation  between  the  parameters  P 
and  R  in  the  vertical  equation  to  the  Conic.  —  From  the 
first  of  the  equations  at  (A)  in  Art.  663,  we  have 


(1). 
Also,  by  combining  both  of  the  equations  at  (A), 

(2). 

II.   To  determine  the  axes  and  eccentricity  of  a  conic 
given  by  the  general  equation.  —  Comparing  (3)  of  Art. 


506  ANALYTIC  GEOMETRY. 

663  with  (3),  (<f),  and  (e)  of  Art.  156,  and  taking  the 
radicals  in  (d)  and  (e)  as  negative,  we  get   . 


A!  _A+B—Q 

C1 


1f~ 


where   Q2  =  (A  —  B)2  +  (2#)2,  and  C"  [Art.  155, 
=  —J:(H2  —  AB)  .     Hence, 

2J 


(H*  —  AB)  (A  +  B  +  Q) 

These  equations  give  the  semi-axes  in  terms  of  the 
general  co-efficients.  For  the  eccentricity,  we  have,  by 
putting  e2  —  1  for  R  in  (2)  above, 


-  9   -  9 

a2  a2 

Substituting  for  a2  and  b2  from  (3),  we  therefore  get 


669.  Two  conies   that  have  the  same   eccentricity, 
are  said  to  be  similar.     It  follows,  then,  that  all  circles, 
all  parabolas,   arid    (Art.   540)   all  hyperbolas   included 
within  equally  inclined  asymptotes,  are  similar. 

Moreover,  since  e  (Art.  668)  is  a  function  of  A,  B,  H, 
these  co-efficients  must  be  the  same  for  all  similar  conies. 

THE  CONIC  IN  THE  ABRIDGED  NOTATION. 

670.  The    Auliarmoiiic    Ratio. — With   respect   to 
this  ratio,  we  shall  only  develop  the  fundamental  prop- 


ANHARMONIC  PROPERTY  OF  CONICS.  ^  507 

erty  of  the  Conic.  The  reader  Ayho  desires  to  follow 
this  property  through  its  manifold  consequences,  may 
consult  the  writings  of  Salmon  and  Chasles. 

Let  A,  B,  (7,  D  be  four  fixed  points  on  any  conic, 
and  0  the  variable  point  of  the  curve.  Then,  if  «,  ^9,  f,  d 
be  the  equations  to  the  four  chords  which  connect  the 
fixed  points,  the  equation  to  the  curve,  referred  to  this 
inscribed  quadrilateral  (see  paragraph  2d,  p.  286)  will 
be 

(1). 


Now,  if  a,  6,  c,  d  denote  the  lengths  of  the  four  chords, 
we  have,  for  the  lengths  of  the  perpendiculars  let  fall 
upon  the  chords  from  0, 

OA.  OB  sin  AOB  OB.OCsmBOC 

—    '->:**•      —    -' 

OC.ODsmCOD  OD.OAsmDOA 


. 

c  d 

Substituting  in  (1),  and  reducing, 

sin  A  OB  sin  COD  _  ,  a  .  c  ,~\ 

= 


But  (Art.  285)  the  first  member  of  (2)  is  the  anharmonic 
of  the  pencil  0-ABCD,  and  the  second  is  constant. 
Hence,  The  anharmonic  of  a  pencil  radiating  from  any 
point  of  a  conic  to  four  fixed  points  of  the  curve  is 
constant. 

671.  Definitions.  —  In  any  hexagon,  two  vertices  are 
said  to  be  opposite,  when  they  are  separated  by  two 
others.  Thus,  if  J.,  _#,  (7,  D,  E,  F  are  the  six  successive 
vertices,  A  and  _D,  B  and  E,  C  and  F  are  opposite. 

Two  sides  are  also  called  opposite  when  separated  by 


508  ANALYTIC  GEOMETRY. 

two  others.  Thus,  AB  and  DE,  BC  and  EF,  CD  and 
FA  are  opposite  sides. 

Opposite  diagonals  are  those  which  join  opposite  ver- 
tices, and  are  therefore  three  in  number ;  namely,  AD, 
BE,  OF. 

672.  Pascal's  Theorem. — Let  0,  /3,  f,  X,  /*,  v  be  the 

successive  sides  of  a  hexagon  inscribed  in  any  conic. 
Then,  if  d  be  the  diagonal  joining  the  opposite  vertices 
va  and  y/,  the  equations 

ar  —  fc/fcj  --  0,     fa  —IfjLd  =  0  (1) 

will  each  represent  the  conic.  We  may  therefore  sup- 
pose the  constants  k  and  I  to  be  so  taken  that  07- —  kfto 
is  identically  equal  to  fa  —  l[w;  that  is,  in  such  a  manner 
that 

ar  —  h  =  (kp  —  l/Ji)d  (2). 

Hence,  all  the  conditions  that  will  cause  07 — fa  to  vanish 
identically,  are  included  in 

3  =  0,     kfi  —  lp  =  0  (3). 

Now  the  points  v0,  yA  evidently  satisfy  a.f — fa  =  0,  and 
these  by  hypothesis  are  on  the  line  d  =  0 :  so  that  the 
points  ax,  j-v,  which  also  satisfy  07-  —  fa  =  0,  but  which 
by  hypothesis  are  not  on  the  line  d  —  0,  must  lie  upon 
the  line  &/?  —  1/2  =  0.  But,  by  the  form  of  its  equation, 
this  line  contains  the  point  p[j>.  In  short,  0.x,  y9//,  ^v, 
which  are  the  intersections  of  the  opposite  sides  of  the 
hexagon,  are  all  on  the  same  line.  Hence,  The  opposite 
sides  of  any  hexagon  inscribed  in  a  conic  intersect  in  three 
points  which  lie  on  one  right  line. 

This  is  known  as  Pascal's  Theorem.  From  this  single 
property,  its  discoverer  BLAISE  PASCAL  is  said  to  have 
developed  the  entire  doctrine  of  the  Conic,  in  a  system 


THEOREMS  OF  PASCAL  AND  BRIANCHON.       509 

of  four  hundred  theorems,  when  he  was  but  sixteen  years 
old;  but  his  treatise  was  never  published,  and  has  un- 
fortunately been  lost.  Leibnitz,  however,  has  given  a 
sketch  of  it,  in  a  letter  written  in  1676  to  Pascal's 
nephew  Perier. 

By  joining  six  points  on  a  conic  in  every  possible  way, 
we  can  form  sixty  different  figures,  each  of  which  may  be 
called  an  inscribed  hexagon,  and  in  each  of  which  the 
intersections  of  the  opposite  sides  will  lie  on  one  right 
line.  Consequently  there  are  sixty  such  lines  for  every 
six  points  on  the  curve,  which  are  called  the  Pascal  lines, 
or  simply  the  Pascals,  of  the  corresponding  conic. 

O73.  Brianciaon's  Theorem. — If  we  take  the  sym- 
bols of  the  preceding  article  as  tangentials,  «,  /9, 7-,  X,  /;.,  v 
will  be  the  vertices  of  a  hexagon  circumscribed  about  a 
conic,  and  3  will  denote  the  intersection  of  the  opposite 
sides  va,  fL  The  equations  at  (1)  will  then  be  tangen- 
tial equations  to  the  conic,  and  the  relation  (2)  will  show 
that  the  three  lines  aX,  /?//,  ?v  intersect  in  the  same  point 
Jcft  — 1/2  =  0.  That  is,  The  three  opposite  diagonals  of  any 
hexagon  circumscribed  about  a  conic  meet  in  one  point. 

This  is  known  as  Brianchon's  Theorem,  having  been 
discovered  in  the  early  part  of  the  present  century  by 
BRIANCHON,  a  pupil  of  the  Polytechnic  School  of  Paris. 
It  was  one  of  the  fruits  of  Poncelet's  Method  of  Recip- 
rocal Polar  s. 

By  producing  six  tangents  to  a  conic  till  they  meet 
in  every  possible  way,  we  can  form  sixty  different  figures, 
each  of  which  may  be  called  a  circumscribed  hexagon. 
Consequently,  for  every  six  points  of  a  conic,  there  are 
sixty  different  Brianchon  points,  determined  by  the 
system  of  six  tangents;  just  as  there  are  sixty  different 
Pascal  lines,  determined  by  the  system  of  six  chords. 


510  ANALYTIC  GEOMETRY. 

EXAMPLES  ON  THE   CONIC  IN  GENERAL. 

1.  If  two  chords  at  right  angles  to  each  other  be  drawn  through 
a  fixed  point  to  meet  any  come,  to  prove  that 


=  constant' 


where  $,  s  are  the  segments  of  one  chord,  and  /S*7,  s/  the  segments 
of  the  other. 

2.  If  through  a  fixed  point  O  there  be  drawn  two  chords  to  any 
conic,  and  if  their  extremities  be  joined  both  directly  and  trans- 
versely, to  prove  that  the  line  PQ  which  joins  the  intersection  of 
the  direct  lines  of  union  to  the  intersection  of  the  transverse  ones 
is  the  polar  of  O. 

3.  Prove  that  any  right  line  drawn  through  a  given  point  to 
meet  a  conic,  is  cut  harmonically  by  the  point,  the  curve,  and  the 
polar  of  the  point;    also,  that  the  chord  through  any  given  point, 
and  the  line  which  joins  that  point  to  the  pole  of  the  chord,  are 
harmonically  conjugate  to  the  two  tangents  drawn  from  the  point. 

4.  A  conic  touches  two  given  right  lines  :  to  prove  that  the  locus 
of  its  center  is  the  right  line  which  joins  the  intersection  of  the 
tangents  with  the  middle  point  of  their  chord  of  contact. 

5.  Prove    that   in   any    quadrilateral    inscribed    in    a    conic,  as 
ABCD,  either  of  the  three  points  E,  F,  O 

is  the  pole  of  the  line  which  joins  the 
other  two.  By  means  of  this  property, 
show  how  to  draw  a  tangent  to  any  conic 
from  a  given  point  outside,  with  the  help 
of  the  ruler  only. 

[This  graphic  problem  is  only  one  of  a 
series  resulting  from  the  method  of  transversals  and  anharmonics,  all  of 
which  are  solvable  with  the  ruler  alone  :  for  which  reason>  the  doctrine 
of  the  solutions  is  sometimes  called  Lineal  Geometry.'] 

6.  Prove  that  in  any  quadrilateral  circumscribed  about  a  conic, 
each  diagonal  is  the  polar  of  the  intersection  of  the  other  two. 


BOOK    SECOND: 

CO-ORDINATES  IN  SPACE. 


An.  Ge.  46.  (511) 


CO-ORDINATES  IN  SPACE. 


674.  In  removing,  at  this  point  in  our  investigations, 
the  restriction  which  has  confined  loci  to  a  given  plane, 
we  shall  only  enter  upon  the  consideration  of  the  most 
elementary  parts  of  the  Geometry  of  Three  Dimensions. 
That  is,  we  shall  only  undertake  to  give  the  student  a 
clear  general  outline  of  the  principles  by  which  we  rep- 
resent and  discuss  the  surfaces  of  the  First  and  Second 
orders.  In  order  to  accomplish  this,  we  must  begin,  as 
in  the  case  of  the  Geometry  of  Two  Dimensions,  by 
explaining  the  conventions  for  representing  a  point  in 
space. 


CHAPTER   FIRST. 
THE    POINT. 

675.  About  a  century  after  the  publication  of  Des- 
cartes' method  of  representing  and  discussing  plane 
curves,  CLAIRAUT  extended  the  method  to  lines  and 
surfaces  in  space,  by  the  following  contrivance  for 
representing  the  position  of  any  conceivable  point  in 
space. 

(513) 


514 


ANALYTIC  GEOMETRY. 


Let  XY,  YZ,  ZX  be  three  planes  of  indefinite  extent, 
intersecting  each  other 
two  and  two  in  the  lines 
X'X,  Y'Y,  Z'Z.  [The 
point  Yf  is  supposed  to 
be  concealed  behind  the 
plane  ZX  in  the  dia- 
gram.] Then,  if  P  be 
any  point  whatever  in 
the  surrounding  space, 
its  position  will  be  known 
with  reference  to  the  three 
planes  so  soon  as  we  find 

the  length  of  PM  drawn  parallel  to  OZ,  and  of  ML,  MN 
drawn  parallel  respectively  to  OF  and  OX-,  or,  which  is 
obviously  the  same  thing,  so  soon  as  we  find  the  lengths 
of  OL,  LM,  MP.  In  the  diagram,  the  three  planes  are 
represented  at  right  angles  to  each  other :  a  restriction 
which  has  the  advantage  of  simplifying  the  whole  subject, 
and  which  can  always  be  secured  by  a  proper  transfor- 
mation, if  the  planes  are  in  fact  inclined  at  any  other 
angle.  We  shall  therefore  suppose,  in  our  investigations, 
that  these  reference-planes  are  always  rectangular,  unless 
the  contrary  is  stated. 

The  distances  OL,  LM,  MP,  or  their  equals  OL,  ON, 
OS,  are  called  the  rectangular  co-ordinates  of  P,  and  are 
respectively  represented  by  x,  y,  z.  The  lines  OX,  OY, 
OZ,  of  indefinite  extent,  are  termed  the  axes :  OX  is  the 
axis  of  x,  OY  the  axis  of  y,  and  OZ  the  axis  of  z.  The 
point  0,  in  which  the  three  axes  intersect,  and  which  is 
therefore  common  to  the  three  reference-planes,  is  named 
the  origin. 

The  reference-planes  evidently  divide  the  surrounding 
space  into  eight  solid  angles,  which  are  numbered  as 


RECTANGULAR  CO-ORDINATES  IN  SPACE.     515 

follows:  Z-XOY  is  the  first  angle;  Z-  YOX',  the 
second;  Z-X'OY',  the  third;  and  Z-Y'OX,  the  fourth. 
Similarly,  Zf - XOY  is  the  fifth  angle;  Z'-YOX',  the 
8ta/i ;  Z'-X'OY',  the  semi/A;  and  Z'-Y'OX,  the 
eighth. 

By  affecting  the  co-ordinates  a:,  ?/,  2  with  the  proper 
sign,  we  represent  a  point  in  either  of  the  eight  angles. 
Thus, 

First   angle  :  x  —  -fa,  y  =  +  J,  2  =  +  <? 

Second    "        a;  =  —  a,  y  =  -f  6,  2  =  -f  c 

Third       "       z  =  —  a,  y  =  —  b,  z  =  +  c 

Fourth     "       x  =  -}-  a,  y  —  —  6,  z  =  -\-  c 

Fifth        «        <&  ==  +  a,  y  =  +  6,  z  =  —  c 

Sixth  "  2:  =  «,       y   rrr    -|-    £>9       g   =  £ 

Seventh  "       x  =  —  a,    t/  =  —  6,    2;  —  —  c 
Eighth     "        x  =  •  -f  a,    y  —  —  6,    z  —  —  c. 

The  student  will  observe  that  the  positive  x  lies  to  the 
right  of  the  first  vertical  plane  YZ,  and  the  negative  x 
to  the  left  of  that  plane ;  the  positive  y,  in  front  of  the 
second  vertical  plane  ZX,  and  the  negative  y  in  the  rear 
of  that  plane ;  the  positive  2,  above  the  horizontal  plane 
JTF,  and  the  negative  z  below  that  plane. 

Corollary  1. — For  any  point  in  the  plane  YZ,  we  shall 
evidently  have 

x  =  0  (1), 

while  y  and  z  are  indeterminate.     Equation  (1)  is  there- 
fore the  equation  to  the  first  vertical  reference-plane. 
For  any  point  in  the  plane  ZX,  we  shall  have 

y  =  o  (2), 

while  z  and  x  are  indeterminate.  Hence,  (2)  is  the 
equation  to  the  second  vertical  reference-plane. 


516  ANALYTIC  GEOMETRY. 

Finally,  for  any  point  in  the  plane  XY,  we  shall 
have 

z  =  0  (3), 

while   x  and  y  are  indeterminate.     Hence,   (3)  is  the 
equation  to  the  horizontal  reference-plane. 

Corollary  2, — If  a  point  is  on  the  axis  X'X,  we  shall 
have  y  =  0,  z  =  0  simultaneously,  while  x  is  indetermi- 
nate. If  the  point  is  on  the  axis  Y'  Y,  z  =  0,  x  =  0 
simultaneously,  while  y  is  indeterminate.  If  the  point 
is  on  the  axis  ZrZ,  x  =  Q9  y  =  0  simultaneously,  while 
z  is  indeterminate.  Hence,  the  pairs 

3r  =  OV         z=Q\         x  =  0 

z  =  0/?         *=0|  y  =  0 

are  respectively  the  equations  to  the  axis  of  x,  the  axis 
of  y,  awcZ  the  axis  of  z. 

Corollary  3. — At  the  point  O,  where  the  axes  intersect 
each  other,  we  shall  evidently  have,  simultaneously, 

x  =  y  =  z  =  0, 
and  these  three  equations  are  the  symbol  of  the  origin. 

POLAR  CO-ORDINATES  IN  SPACE. 

676.  If  MN  be  a  fixed  plane,  OX  a  fixed  line  in  it, 
and  0  a  fixed  point  in  that 
line,  then,  if  any  point  P  in 
the  surrounding  space  be 
joined  with  0,  and  a  plane 
be  passed  through  OP  per- 
pendicular to  MN,  so  as  to 
intersect  the  latter  in  the 
line  OR,  the  distance  OP,  and  the  angles  FOR,  ROX, 


POLAR  CO-ORDINATES  IN  SPACE.  517 

are  called  the  polar  co-ordinates  of  the  point  P.  The 
distance  OP  is  called  the  radius  vector,  and  is  repre- 
sented by  the  letter  p;  the  angles  POR,  ROX  are 
termed  the  vectorial  angles,  and  are  designated  respect- 
ively by  <p  and  0,  as  in  the  diagram. 

MN  is  called  the  initial  plane,  OX  the  initial  line, 
and  0  the  pole.  Instead  of  the  angle  y,  its  complement 
is  sometimes  used,  designated  by  f. 

By  inspecting  the  diagram,  it  will  be  evident  that  we 
may  use 

?  =  o  (i) 

as  the  equation  to  the  initial  plane, 

<p  =  6  =  0  (2) 

as  the  equations  to  the  initial  line,  and 

P  =  0  .          (3) 

as  the  equation  to  the  pole. 

THE  DOCTRINE  OF  PROJECTIONS. 

677.  Definitions, — The  point  in  which  a  line  in  space 
pierces  a  given  plane,  is  called  the  trace  of  the  line  upon 
the  plane.  Similarly,  the  line  in  which  a  surface  cuts  a 
given  plane,  is  termed  the  trace  of  the  surface  upon  the 
plane.  In  particular,  the  trace  of  one  plane  upon  an- 
other, is  the  right  line  in  which  the  former  intersects  the 
latter. 

If  a  perpendicular  be  let  fall  from  any  point  to  a 
given  plane,  the  trace  of  the  perpendicular  upon  the 
plane  is  called  the  orthogonal  projection  of  the  point  on 
the  plane.  When  we  use  the  term  projection  in  what 


518 


ANALYTIC  GEOMETRY. 


follows,  we  shall  always  intend  an  orthogonal  projection. 
Thus,  in  particular,  the 
projections  of  a  point  P 
on  the  three  reference- 
planes,  are  respectively 
M7  R,  S,  the  traces  of  its 
three  co-ordinates. 

The  projection  of  any 
curve  upon  a  given  plane, 
is  the  curve  formed  by 
projecting  all  of  its  points. 
The  perpendiculars  let 
fall  in  forming  such  a 

projection  will  of  course  form  a  surface,  which  is  called 
the  projecting  cylinder  of  the  curve. 

When  the  curve  projected  is  a  right  line,  it  is  obvious 
that  the  projecting  cylinder  will  become  a  plane.  Hence, 
the  projection  of  any  right  line  upon  a  given  plane  is 
the  right  line  in  which 
the  projecting  plane  cuts 
the  given  plane.  For 
example,  the  projection 
of  the  radius  vector  OP 
upon  the  initial  plane  MN, 
is  the  line  OR. 

The  projection  of  a  point  upon  a  given  line,  is  the 
trace  of  that  line  upon  the  plane  which  passes  through 
the  given  point  and  is  perpendicular  to  the  given  line. 
Thus,  L,  N9  Q  are  the  projections  of  a  point  P  upon  the 
three  co-ordinate  axes. 

The  projection  of  a  right  line  upon  a  given  one,  is  the 
portion  of  the  latter  included  between  the  projections  of 
the  extremities  of  the  former.  For  example,  OL  is  the 
projection  of  RP  on  the  axis  of  x. 


ORTHOGONAL  PROJECTIONS. 


519 


The  angle  which  any  right  line  makes  with  a  given 
plane,  *is  the  angle  included  between  the  line  and  its 
projection  on  the  plane;  the  angle  which  it  makes  with 
a  given  line,  is  the  angle  included  between  it  and  an 
intersecting  parallel  to  that  line. 

678.  Theorem. —  The  projection  of  .a  finite  right  line 
upon  any  plane  is  equal  in  length  to  the  length  of  the  line 
multiplied  by  the  cosine  of  the  angle  between  the  line  and 
the  plane. 

Let  XY  be  the  given  plane, 
and  PQ  the  given  line.  Then, 
if  M  and  N  be  the  projections 
of  P  and  Q,  the  projection  of 
PQ  will  be  MN.  Now,  drawing 
PR  parallel  to  MN,  we  get 
(Trig.,  858) 

MN=  PR  =  PQ  cos  QPR: 
which  proves  our  proposition. 

67O.  Theorem. —  The  projection  of  one  finite  right  line 
upon  another  is  equal  in  length  to  the  length  of  the  first 
multiplied  by  the  cosine  of  the  angle  between  the  two.. 

Let  the  first  line  be  PQ,  and  the  second  OX.  Then, 
if  we  pass  through  P  and  Q  the  planes  PTL,  QIF,  per- 
pendicular to  OX,  the  projection  of  PQ  upon  OX  will 
be  IL.  Let  PF  now  be  drawn  parallel  to  OX:  it  will 
be  perpendicular  to  the  plane  QTF  at  F.  Then,  in  the 
right-angled  triangle  QPF,  PF  =  PQ  cos  QPF.  But, 
by  the  construction,  PF  =  IL.  Hence, 

IL  =  PQ  cos  QPF: 


which  proves  the  proposition. 
An.  Ge.  47. 


520 


ANALYTIC  GEOMETRY. 


68O.  Theorem. — In  any  series  of  points,  the  projection 
(on  a  given  line)  of  the  line  which  joins  the  first  and  last, 
is  equal  to  the  sum  of  the  projections  of  the  lines  which  join 
the  points  two  and  two. 

The  points  may  be  so  situated  that  their  projections  on 
the  given  line  advance  successively  from  the  first  to  the 
last :  in  which  case  the  theorem  is  an  obvious  consequence 
of  the  sixth  definition  in  Art.  677.  Or  they  may  be  so 
placed  that  the  projections  of  some  fall  on  the  given  line 
behind  those  of  the  next  preceding  points :  in  which  case 
we  still  obtain  the  theorem,  if  we  consider  the  line  which 
joins  such  a  point  to  its  predecessor  as  forming  a  negative 
projection,  and  understand  the  sum  mentioned  above  as 
algebraic. 

Corollary. —  The  projection  of  the  radius  vector  of  any 
point,  is  equal  to  the  sum  of  the  projections  of  the  co- 
ordinates of  the  point. 

For  the  points  0,  L,  M,  P  (see  diagram,  Art.  675) 
may  of  course  be  considered  as  a  series  coming  under 
the  above  theorem. 


DISTANCE  BETWEEN  TWO  POINTS  IN  SPACE. 

681.  Let  P  and  Q  be  the  two  points,  projected  re- 
spectively at  M,  R,  S  and 
JV,  T,  V.  Through  P,  pass  a 
plane  RPF,  parallel  to  the  ref- 
erence-plane XY\  and  \QiPML 
be  the  projecting  plane  of  MP, 
and  QNH  of  NQ.  Then,  in  the 
right-angled  triangle  QPF,  we 
shall  have 


PQf  = 


(I)- 


DIRECTION-COSINES  OF  A  LINE.  521 

But,  by  the  construction  of  the  figure,  PF=MN',  hence, 
from  the  right-angled  triangle 


PF2  =  NG2  +  MG2  (2). 

Substituting  in  (1),  we  obtain 

PQ2  =  NG2  +  MG2  +  QF2. 

Hence,  if  the  co-ordinates  of  P  be  a/,  ?/',  z',  and  those 
of  Q  be  x",  y",  z",  while  d  represents  the  distance  PQ, 
we  have 

d2  =  0"  —  xj  +  (y"  —  yj  +  (*"  —  O2. 

Corollary.  —  For  the  distance  from  the  origin  to  any 
point  xy  in  space,  we  therefore  have  (Art.  675,  Cor.  3) 


The  last  result  may  be  interpreted  thus:  The 
square  on  the  radius  vector  of  any  point  is  equal  to  the 
sum  of  the  squares  on  the  co-ordinates  of  the  point. 

This  theorem  leads  to  a  remarkable  relation  among  the 
so-called  direction-cosines  of  a  right  line,  that  is,  the  co- 
sines of  the  three  angles  which  the  line  makes  with  the 
three  co-ordinate  axes.  Let  the  angle  made  with  the 
axis  of  x  be  «,  that  made  with  the  axis  of  y  be  /?,  and 
that  made  with  the  axis  of  z  be  f.  Then,  supposing  a 
parallel  to  the  given  line  to  be  drawn  through  the  origin, 
the  co-ordinates  of  any  point  xyz  on  this  parallel  will  be 
the  projections  of  its  radius  vector  on  the  axes,  and  we 
shall  have  (Art.  679) 

x  =  p  cos  a,      y  =  f>  cos  /9,      z  =  p  cos  ^. 

Squaring  and  adding  these  equations,  and  observing  that 
p2  =  x2  -f-  y2  -f-  z2,  we  get,  for  the  relation  mentioned, 

cos2  a  -f  cos2  ft  +  cos2  f  =  1. 


522 


ANALYTIC  GEOMETRY. 


POINT  DIVIDING  THE  DISTANCE  BETWEEN  TWO  OTHERS 
IN  A  GIVEN  RATIO. 

683.  By  an  investigation  analogous  to  that  of  Art.  52, 
the  details  of  which  the  student  can  easily  supply,  the  co- 
ordinates of  such  a  point  are  found  to  be 


mx2  -j- 


m  -j-  n 


= 
J~ 


™#j  +  ny\ 


z  = 


mz. 


m  -\-  n 


TRANSFORMATION  OF  CO-ORDINATES. 

684.    To  transform  to  parallel  reference  -planes  passing 
through  a  new  origin. 

Let  xf9  yfj  z'  be  the  co-or- 
dinates of  the  new  origin,  #, 

o       "       " 

i/,  z  the  primitive  co-ordinates 
of  any  point  P,  and  X,  Y,  Z 
its  co-ordinates  in  the  new 
system.  Then,  as  is  evident 
upon  inspecting  the  diagram, 
the  formulae  of  transformation 
will  be 


685.    To  transform  from  a  given  rectangular  system  to 
a  system  having  its  planes 
at  any  inclination. 

Let  the  direction-angles 
of  the  new  axis  of  x  be  «,  ft, 
Y  ;  those  of  the  new  axis  of 
T/,  a',  /9r,  f  ;  and  those  of  the 
new  axis  of  2,  a",  /9",  f. 
Then,  if  we  suppose  each  of 
the  new  co-ordinates  N'P, 
M'P,  Q'P  to  be  projected 


TRANSFORMATION  OF  CO-ORDINATES. 


523 


on  one  of  the  old  axes,  the  sum  of  the  three  projections 
(Art.  680,  Cor.)  will  in  each  case  be  equal  to  the  projec- 
tion of  the  radius  vector  OP.  But  the  projection  of  OP 
on  OX  will  be  equal  to  the  old  x  of  P;  its  projection  on 
OF,  to  the  old  y\  and  its  projection  on  OZ,  to  the  old  z. 
Hence,  (Art.  679,) 

x  =  X  cos  a  -f  Y  cos  a!  -f-  Z  cos  a!', 
y  =  A"cos/9  +  Fcos/5'  +  Zcos/?", 
2  =  A7"  cos  f  •-}-  F  cos  f'  -f  Z  cos  f", 

are  the  required  formulae  of  transformation. 

Remark, — It  must  be  borne  in  mind,  in  using  these 
formulae,  that  the  direction-cosines  of  the  new  axes  are 
subject  to  the  conditions  (Art.  682) 

cos2  a  -\-  cos2  /9  +  cos2  f  =  1, 
cos2  a'  -j-  cos2  ft  +  cos2  f  =  1, 
cos2  a"  -j-  cos2/3"  +  cos2  fr  =  I. 

6860  To  transform  from  a  planar  to  a  polar  system 
in  space. 

Let  the  planar  system  be  rectangular.     Then,  the  co- 
ordinates of  any  point  P  in  the 
two  systems  being  related  as  in 
the  diagram,  it  is  evident  that 
we  shall  have 


x  =  p  cos 
y  =  p  cos 
z  =  sin 


cos    , 
sin  6, 


From  these  equations  we  can  evidently  also  find  />,^>, 
in  terms  of  x,  y,  z. 


524  ANALYTIC  GEOMETRY. 

Remark. — To  combine  a  change  of  origin  with  this 
transformation  or  that  of  the  preceding  article,  we  have 
merely  to  add  the  co-ordinates  a/,  y' ,  z'  of  the  new  origin 
to  the  values  found  for  x,  y,  and  z. 

GENERAL  PRINCIPLES  OF  INTERPRETATION. 

687.  These  follow  from  the  convention  of  co-ordinates 
in  space  in  much  the  same  manner  as  the  principles  of 
Plane  Analytic  Geometry  followed  from  the  convention 
of  plane  co-ordinates.  We  may  therefore  state  them 
without  further  argument,  as  follows: 

I.  Any  single  equation  in  space-coordinates  represents 
a  surface. 

To  hold  with  full  generality,  this  statement  must  be 
understood  to  include  (in  addition  to  surfaces  in  the 
ordinary  sense)  imaginary  surfaces,  surfaces  at  infinity, 
surfaces  that  have  degenerated  into  lines  or  points,  and 
surfaces  combined  in  groups. 

II.  Two   simultaneous   equations   in   space-coordinates 
represent  a  line  of  section  between  two  surfaces. 

This  principle  is  also  to  be  taken  with  restrictions 
corresponding  to  those  above  stated. 

III.  Three  simultaneous  equations  in  space-coordinates 
represent  mnp  determinate  points. 

These  are  the  points  of  intersection  of  three  surfaces, 
supposed  to  be  of  the  wth,  wth,  and  pih  order  respectively. 

IV.  An  equation  which  lacks  the  absolute  term,  repre- 
sents a  surface  passing  through  the  origin. 

V.  Transformation  of  co-ordinates  in  space  does  not 
alter  the  degree  of  a  given  equation,  nor  affect  the  form 
of  its  locus  in  any  way. 


THE  PLANE.  525 


CHAPTER   SECOND. 
LOCUS  OF  THE   FIRST  ORDER   IN  SPACE. 

688.  Form  of  the  Locus. — The  general  equation  of 
the  first  degree  in  three  variables,  may  be  written 

Ax+By+Cz+D  =  Q  (1), 

where  A,  J9,  (7,  D  are  any  four  constants  whatever. 

Transforming  (1)  to  parallel  axes  passing  through  a 
new  origin  x'y',  we  get  (Art.  684) 

Ax  +  By  +  Cz  +  (Ax'  +  BiJ  +  Czr  +  D)  =  0. 

Hence,  if  we  suppose  the  new  origin  to  be  any  fixed 
point  in  the  locus  of  (1),  the  new  absolute  term  will 
vanish,  and  our  equation  will  take  the  form 

Ax  +  JBy+Cz  =  Q  (2). 

If  we  now  change  the  directions  of  the  reference- 
planes  (Art.  685),  we  shall  get,  after  expanding  and 
collecting  terms, 

(A  cos  a    -{•  B  cos  /?    -f-  C  cos  7-  )  x  ^ 

4-  (A  cos  af  +  B  cos  ft'  +Ccosr')y  >  =  0. 
+  (A  cos  a!'  +  B  cos  /3"  +  (7 cos  r"}  z  ) 

Hence,  if  we  can  take  the  new  reference-planes  so  as  to 
give  the  new  axis  of  x  and  the  new  axis  of  y  such 
directions  that 

A  cos  a  -J-  B  cos  /9  -f-  C  cos  y  =  0 

==  A  cos  a!  +  £  cos  /9'  +  (7  cos  /         (Q), 

we  shall  reduce  our  equation  to  the  simple  form 

Z  =  0  (3). 


526  ANALYTIC  GEOMETRY. 

Now,  obviously,  the  transformation  from  (1)  to  (2)  is 
always  possible ;  and  that  we  can  always  effect  the 
transformation  from  (2)  to  (8)  will  readily  appear. 
For  Ave  can  leave  the  primitive  vertical  reference-planes 
unchanged,  obtaining  our  new  system  by  merely  revolv- 
ing the  primitive  horizontal  plane  about  the  origin:  in 
which  case,  we  shall  have 

«  =  90°  —  r,     0  =  0;     a'  =  0,     /S'^OO0  —  rf ; 

and  the  conditions  at  (Q),  upon  which  the  transformation 
we  are  now  considering  depends,  will  become 

C 

A  sin  f  -f  C  cos  ?  =  0    i.  e.   tan  ^  — _  , 

J*. 

B  sin/  -f  C  cos/  =  0    i.e.   tan/  =  _^: 

JD 

suppositions  compatible  with  any  real  values  of  J.,  £,  C. 
We  conclude,  then,  that  by  a  proper  transformation 
of  co-ordinates  we  can  always  reduce  the  general  equa- 
tion of  the  first  degree  to  the  form 

3  =  0. 

But  this  [Art.  675,  Cor.  1,  (3)]  denotes  the  new  refer- 
ence-plane XY.  Hence,  (Art.  687,  V,)  The  locus  of  the 
First  order  in  space  is  the  Plane;  or,  as  we  may  otherwise 
state  our  result,  Every  equation  of  the  first  degree  in  space 
represents  a  plane. 

THE  PLANE  UNDER  GENERAL  CONDITIONS. 

689.  General  Form  of  ttoe  Equation  to  the 
Plane. — From  what  has  just  been  shown,  we  learn 
that 

Ax  -f  By  -f  Cz  +  D  =  0 

is  the  Equation  to  any  Plane. 


PLANE  UNDER  GENERAL  CONDITIONS.        527 


o  The  Plane  in  terms  of  its  Intercepts  on 
the  Axes.  —  Let  the  plane  ABC,  making  upon  the  co- 
ordinate axes  the  intercepts  OA  =  a,  OB  —  b,  OC=c, 
represent  any  plane 


Ax  -f  By  +  Cz  +  D  =  0. 

Making  ?/  and  0,  2-  and  x,  x  and  ^ 
simultaneously  =  0  in  succession, 
we  obtain  from  this  equation  (Art. 
675,  Cor.  2) 

D 

x=  a  = r     .-. 

A 

D 

y  =  t=--  .-. 


z  =  c  =  — 


D 


Substituting  these  values  of  A9  B,  C  in  the  general 
equation,  we  obtain  the  equation  to  the  Plane  in  terms 
of  its  intercepts,  namely, 


The  Plane  in  terms  of  the  Direction-cosines 
of  its  Perpendicular. — Let  the  perpendicular  from  the 
origin  upon  any  plane  be  =  p,  and  let  its  direction-angles 
be  a,  /?,  f.  Then,  a,  6,  c  being  the  intercepts  of  the  plane, 
we  shall  have 


cos  a. 


cos 


COSf 


528  ANALYTIC  GEOMETRY. 

Substituting  these  values  in  the  equation  of  the  preced- 
ing article,  we  obtain 

x  cos  o.-\-y  cos /? -f-  z  cos  f—p, 
the  equation  to  the  Plane  in  the  terms  now  required. 

692.  Reduction  of  the  General  Equation  to  the 
form  last  found. — We  may  suppose  the  reduction  to  be 
effected  by  dividing  the  general  equation 

Ax  +  By  +  Cz  +  D  =  0 

throughout  by  some  quantity  Q.     If  so,  we  shall  have 
A  =  Qcosa,  .5=^()cos/3,   C=Qcosf:  whence 

Q1  (cos2  a  +  cos2  /9  4-  cos2  r)  =  A2  +  B2  -j-  <72. 

Now  (Art.  682),  cos2  a  -f  cos2^  +  cos2  f  =  1.     Hence, 
1- ^  +  6";  and  we  learn  that 


cos  «== 


COS        -=: 


(7 

~ 


and  that,  for  the  perpendicular  from  the  origin  upon  a 
plane  given  by  the  general  equation,  we  have 


D 

P=  — 


By  always  taking  the  radical   Q  with  that  sign  which 
will  render  p  positive,  the  resulting  signs  of  cos  «,  cos  /?, 


PLANE  UNDER  SPECIAL  CONDITIONS.          529 

cos  f  will  indicate  whether  the   direction-angles  of  the 
perpendicular  are  acute  or  obtuse. 

THE  PLANE   UNDER  SPECIAL  CONDITIONS. 

693.  Equation  to  a  Plane  passing  through  Three 
Fixed  Points.  —  By  a  process  exactly  analogous  to  that 
of  Art.  95,  this  is  found  to  be 

C     (y"  «'"-y"V  )*' 

-j  +  (y"V   -y'    z'")x" 

(.  +  (y'  *"  -y"  »'  )*"', 

in  which  x'y'z',  xny"z",  xtrry'"z"f  are  the  three  points 
which  determine  the  plane. 

694.  Angle  between  two  Planes.  —  This  is  evidently 
equal,  or  else   supplemental,  to  the  angle  between  the 
perpendiculars  thrown  upon  the  planes  from  the  origin. 
Now,  if  p,  p'  be   the  lengths   of  these   perpendiculars, 
«,  /9,  7-  and  «',  /3',  f  their  direction-angles,  and   d  the 
distance  between   the  points  xyz,  x'y'z'  in  which   they 
pierce  their  respective  planes,  we  shall  have  (Art.  681) 


where  <p  ==  the  angle  between  the  planes  or  their  per- 
pendiculars. But  (Art.  681,  Cor.)  p2  =  x2  -j-  y2  -f-  ^2, 
and  p'2  =  x'2  -f  y'2  -f-  z'2.  Hence,  after  obvious  reduc- 
tions, 

ppf  cos  <p  =  xxr  +  ^'  -f  zzf; 


or,  since  a:=^^?cos«,  y=pcosfl,  z=pcosf;  x'=pfcosaf, 

y'  =  pr  cos  /9',  2r  ==  pr  cos  f, 

cos  ^  =  cos  a  cos  «r-j-  cos  /?  cos  ^r-f-  cos  7-  cos  f     (A). 

Here   it  becomes   evident,  upon  a  moment's  reflection, 
that  the  direction-angles  7*  and  f  are  respectively  equal 


530  ANALYTIC   GEOMETRY. 

to  the  angles  which  the  two  planes  make  with  the  hori- 
zontal reference-plane ;  that  a  and  a!  are  respectively 
equal  to  those  made  with  the  first  vertical  plane ;  and 
that  /?  and  pf  are  respectively  equal  to  those  made  with 
the  second  vertical  plane.  Calling  these  new  angles 
£  and  £',  o  and  t/,  £  and  f',  wre  have,  then,  as  the  ex- 
pression for  the  angle  between  two  planes  in  terms  of  their 
inclinations  to  the  reference-planes, 

cos  <p  =  cos  £  cos  £'  -f-  cos  o  cos  o'  -\-  cos  f  cos  f '     (1). 

Replacing  the  cosines  in  (1)  by  their  values  from  Art. 
692,  namely, 

C  C' 

COS  f  =  — 7-T         -TTTx  ,     COS  f'= 


cos  v  =  ———      — -  ,    cos  tr=  - 


COS  C  =  -77-17         ~^r.  ,      COS  C/= 


we  obtain,  as  the  expression  for  the  angle  between  two 
planes  in  terms  of  the  co-efficients  of  their  equations, 

'  AA'+BB'+CCf  ,9s 

~   722       2       '*'*       'z 


Corollary  1.  —  The  two  planes  will  be  parallel  if  <p  —  0  ; 
that  is,  if  cos  <p  =  1.  As  the  condition  of  parallelism, 
then,  the  terms  of  the  second  member  of  (2)  must  be 
equal  ;  or,  after  squaring  and  transposing, 

(AB'  —  A'B)*  4  (BO'  —  B'CY  +  (CAf-C'Ay=,Q  : 

a  condition  which  can  only  be  satisfied  by  having  simul- 
taneously 


PARALLELISM  AND  PERPENDICULARITY.      531 

Corollary  2. — If  the  two  planes  are  perpendicular  to 
each  other,  we  shall  have  cos  <p  =  0 :  whence,  as  the 
condition  of  perpendicularity, 

AAf  +  BB'  +  CC'  =  0. 

695.  Equation  to  a   Plane  parallel  to   a  given 
one. — From  the  condition  reached  in  the  first  corollary 
to  the  preceding  article,  it  is  evident  that  this  can  finally 
be  written  in  the  form 

Ax  +  By  +  Cz  +  D'=Q, 

A,  B,  C  being  the  co-efficients  of  x,  y,  z  in  the  equation 
to  the  given  plane.  We  learn,  then,  that  the  equations 
to  parallel  planes  differ  only  in  their  constant  terms. 

Corollary. — The  equations  to  planes  parallel  respect- 
ively to  the  three  reference-planes,  will  be 

z  =  constant,     x  =  constant,     y  =  constant. 

696.  Equation  to  a  Plane  perpendicular   to  a 
given  one.— If  A'x  +  B'y  -j-  C'z  -f  D1  =  0  be  the  given 
plane,  we  may  write  the  required  equation  in  either  of 
the  forms 

Px  —  By—Cz  —  D  =  Q  (1), 

Ax-Qy+Cz  +  D  =  Q  (2), 

Ax  +  £y  —  Xz+D  =  Q  (3), 

by  merely  making,  in  accordance  with  Art.  694,  Cor.  2, 

_  BB'  +  CO'  CC'+AA'  AA'+BB' 

~7i^          ^  ~W  ~~C1~ 

Corollary. — In  particular,  the  equations  to  planes  per- 
pendicular to  the  reference-planes  will  assume  the  forms 


532  ANALYTIC  GEOMETRY. 

For  (Art.  675,  Cor.  1)  R  must  vanish  for  the  horizontal 
reference  -plane,  P  for  the  first  vertical,  and  Q  for  the 
second. 

697.  ILength  of  the  Perpendicular  from  a  Fixed 
Point  to  a  Given  Plane.  —  Let  the  fixed  point  be  xyz, 
and  the  given  plane  x  cos  a  -f-  y  cos  /9  +  z  cos  7-  —  p  =  0. 
If  we  produce  the  perpendicular  p,  and  then  project  upon 
it  the  radius  vector  of  xyz,  it  is  evident  that  the  required 
perpendicular  will  be  equal  to  the  difference  between  this 
projection  and  p.  Hence,  (Art.  680,  Cor.,)  we  have 

P=±  (xcos  a-\-  y  cos/9  -f-  ^cos^  —  p), 


the  upper  or  lower  sign  being  used  according  as  the 
given  point  and  the  origin  lie  on  opposite  sides  of  the 
given  plane,  or  on  the  same  side. 

Corollary  1.  —  For  the  perpendicular  from  xyz  to  the 
plane  Ax  +  By  -f  Cz  +  D  =  0,  we  have  (Art.  692) 


p       Ax  +  By  +  Cz  -f-  D 
*        n 


Corollary  2.  —  Since  we  have  agreed  to  consider  the 
perpendicular  from  the  origin  upon  any  plane  as  positive 
in  all  cases,  consistency  requires  that  perpendiculars 
dropped  upon  a  plane  from  any  point  on  the  same  side 
of  it  as  the  origin,  shall  be  reckoned  positive;  and  those 
dropped  from  the  opposite  side,  negative. 

698.  Equation  to  a  Plane  passing  through  the 
€0111111011  Section  of  two  given  ones.  —  By  reasoning 
similar  to  that  of  Arts.  107,  108,  it  is  evident  that  this 
may  be  written 

(Ax  +  By  +  Cz  +  D)  +  k  (A'x  +  B'y  +  0  rz  -f-  -#')  =  0  ; 


INTERSECTIONS  OF  PLANES.  533 

or,  by  adopting  abridgments   similar   to   those   used  in 
Plane  Geometry, 

P  +  &P'=0. 

Corollary.  —  Analogy  leads  at  once  to  the  conclusion, 
that  an  equation  of  the  form 


in  which  /,  m,  n  are  arbitrary  constants,  denotes  a  plane 
passing  through  the  point  in  which  the  three  planes  P, 
P',  P"  intersect. 

699.  Equation  to  the  Plane  bisecting  the  angle 
between  two  given  ones.  —  The  reasoning  of  Art.  109 
applies  here,  and  the  required  equation  (Art.  692)  is 

Q'P±QP'  =  Q  (1), 

or,  if  the  equations  to  the  given  planes  are  already  re- 
duced to  terms  of  their  direction-cosines, 

a±fi  =  0  (2), 

the  upper  sign  denoting  the  external  bisector,  and  the 
lower  the  internal  one. 

TOO.  Condition  that  Four  Points  shall  lie  on 
one  Plane.  —  The  fourth  point  must  of  course  satisfy  the 
equation  to  the  plane  of  the  other  three,  and  the  required 
condition  is  therefore  obtained  by  putting  xlvy™zlv  instead 
of  xyz  in  the  equation  of  Art.  693. 

7O1.  Condition  that  Three  Planes  shall  pass 
through  one  Right  Line.  —  The  equation  to  the  third 
plane  must  take  the  form  (Art.  698)  of  the  equation  to 


534  ANALYTIC   GEOMETRY. 

a  plane  passing  through  the  common  section  of  the  other 
two.     There  must,  then,  be  some  constant  —  n,  such  that 


-nP"  =  lP  +  mPf. 
Hence,  the  required  condition  is 


In  other  words,  Three  planes  pass  through  one  right  line 
whenever  their  equations,  upon  being  multiplied  by  three 
suitable  constants  and  added  together,  vanish  identically. 

702.  Condition  that  Four  Planes  shall  meet  in 
One  Point.  —  By  applying  the  reasoning  of  the  preceding 
article  to  the  result  of  the  corollary  to  Art.  698,  we  learn 
that  this  condition  may  be  written 

IP  +  mP1  +  nP"  +  rP'"  =  0  ; 

or,  if  the  equations  to  the  planes  be  in  terms  of  their 
direction-cosines, 

la  +  mp  +  nr  +  rd  =  Q. 

Hence,  Four  planes  pass  through  one  point  whenever  their 
equations,  upon  being  multiplied  by  four  suitable  constants 
and  added  together,  vanish  identically. 

QUADRIPLANAR   CO-ORDINATES. 

703.  The  condition  of  the  preceding  article  subjects 
its  constants  /,  m,  n,  r  to  certain  restrictions,  consistent 
with  the  identical  vanishing  of  the  function 

la  -J-  TTi/9  +  nf  +  rd. 

But  if  we  now  free  these  constants  from  this  condition 
for  the  converging  of  four  planes,  making  them  abso- 


RIGHT  LISE  IN  SPACE.  535 

lately  arbitrary,  we  learn,  by  reasoning  entirely  analo- 
gous to  that  of  Arts.  208—217,  that  if  a  =  0,  /9  =  0, 
Y  _-=  0,  o  =  Q  be  the  equations  to  any  four  planes  forming 
a  tetrahedron,  the  equation 

la  -f-  ???/?  -f-  n?  -f  »*(J  =  0 

is  a  general  symbol  for  any  plane  in  space. 

We  thus  arrive  at  what  may  be  called  a  system  of 
quadriplanar  co-ordinates,  analogous  to  the  trilinear 
system  of  Plane  Geometry. 

LINEAR  LOCI  IN  SPACE. 

704.  By  II  of  Art,  687,  it  appears  that  all  lines  in 
space,  whether  right  or  curved,  are  to  be  solved  as  the 
common  sections  of  two  surfaces,  and  hence  must  be  rep- 
resented by  two  simultaneous  equations  in  three  variables. 
In  particular,  the  Right  Line  in  Space,  which  is  the  only 
line  we  shall  have  room  to  consider,  must  be  treated  as 
the  common  section  of  two  planes. 

705.  Equations  to  the  Right  tine  in  Space.— We 

might  represent  this  line  by  the  two  general  equations 

Ax  +  By  +  Cz+  D=0,     A'x 

but  it  is  far  more  convenient  to  denote  it  by  the  simul- 
taneous equations  of  its  two  projecting  planes  (Art.  677), 
in  accordance  with  the  method  by  which  all  curves  in 
space  are  usually  represented  by  means  of  their  "  pro- 
jecting cylinders." 

In  pursuance  of  this  method,  then,  the  equations  to 
the  Right  Line  projected  upon  the  two  vertical  reference- 
planes,  will  be  of  the  form  (Art.  696,  Cor.) 

By+Cz  +  D=Q,    Nz 
An.  Ge.  48. 


536  ANALYTIC  GEOMETRY. 

Now  it  is  noticeable,  that,  while  these  equations  taken 
together  involve  three  variables,  each  of  them  taken  sep- 
arately involves  but  two.  The  first,  interpreted  as  an 
equation  in  two  variables,  denotes  a  right  line  in  the 
first  vertical  plane;  the  second,  similarly  interpreted, 
denotes  a  right  line  in  the  second  vertical  plane.  But 
these  lines,  by  the  principle  of  the  corollary  to  Art.  696, 
must  also  lie  in  the  two  planes  which  the  equations  denote 
when  interpreted  in  space :  hence,  they  are  the  common 
sections  of  these  planes  and  the  vertical  planes  of  refer- 
ence ;  or,  in  other  words,  they  are  the  projections  of  the 
right  line  represented  by  the  simultaneous  equations 
By  +  Cz  +  D  =  0,  Nz  +  MX  +  L  =  0,  upon  the  two 
.vertical  reference-planes. 

We  see,  then,  that  we  may  either  regard  the  two 
determining  equations  of  the  Right  Line  as  the  space- 
equations  to  its  two  projecting  planes,  or  as  the  plane- 
equations  to  its  two  projections.  It  is  customary  to 
interpret  them  in  the  latter  way,  and  as  each  involves 
two,  and  only  two,  arbitrary  constants,  to  write  them 

x  =  mz  -f-  0,     y  =  nz  -j-  b. 

Thus  the  axis  of  z  is  made  their  common  axis  of  abscissas, 
and  the  constants  m,  a,  n,  b  take  meaning  as  follows  : 
m—  the  tangent  of  the  angle  which  the  projection 

on  the  second  vertical  plane  makes  with  the 

axis  of  z. 
a  =  the  intercept  which  the  same  projection  forms 

on  the  axis  of  x. 
n  =  the  tangent  of  the  angle  which  the  projection 

on  the  first  vertical  plane  makes  with  the 

axis  of  z. 
b  =  the  intercept  which  this  projection  forms   on 

the  axis  of  y. 


EIGHT  LINE  IN  SPACE.  537 

We  learn,  then,  that  the  position  and  direction  of  a  right 
line  in  space,  depend  upon  the  magnitudes  and  signs  of 
four  arbitrary  constants. 

7O6.  Symmetrical  Equations  to  the  Right  IJue 
in  Space.  —  Let  the  line  pass  through  an  arbitrary  point 
x'lj'z',  and  let  its  direction-angles  be  a,  /?,  f. 

Then,  if  I  =  the  distance  from  x'y'z'  to  any  point  xyz 
of  the  line,  the  projections  of  I  upon  the  three  co-ordi- 
nate axes  (Art.  679)  will  be  I  cos  «,  I  cos  /9,  I  cos  7*.  But 
by  definition  (Art.  677)  these  projections  are  respectively 
equal  to  x  —  xf,  y  —  y',  z  —  z'.  Hence, 


a  =  x  —  x1  ',     I  cos  /9  =  y  —  y',     ICQS?  =  Z  —  z'  : 
whence,  solving  for  I  and  equating  the  three  results, 

x  —  x'        y  —  ?/        z  —  z'  % 
cos  a.  cos  /5          cos  y 

which  are  the  symmetrical  equations  sought. 

7O7.  To  find  the  Direction-cosines  of  a  Right 
Line  given  by  its  Projections.  —  The  direction-cosines 
of  any  right  line  are  of  course  the  same  as  those  of  its 
parallel  through  the  origin.  Let  the  projections  of  such 
a  parallel  be 

-  =  V-  =  —  . 
I         m         n 

Then,  if  f>  be  the  radius  vector  of  any  point  xyz  on  the 
parallel,  we  shall  have  (Art.  681,  Cor.) 


and,  from  the  above  equations  of  projection, 
mx  nx 


538  ANALYTIC   GEOMETRY. 

Solving  the  last  three  equations  for  x,  y,  z,  we  obtain 

_  Iff  mp 

=  2        *   '  = 


np 


But,  by  the  doctrine  of  projections, 

x  =  p  cos  «,     y  =  p  cos  /?,     z  =  p  cos  T. 
Substituting,  and  dividing  through  by  p, 

I  n  m 

cos  a  =     — — -  ,     cos  Q  = 


n 


Corollary, — To  find  the  direction-cosines  of  a  line 
whose  projections  are  given  in  any  form  whatever, 
throw  its  equations  into  the  form 

x —  xf y  —  yf z  —  z'  . 

I  m  n 

when  the  required  functions  will  be  Z,  m,  n,  each  divided 

by  vV  H-  wi*  +  w2. 

TO8.  Angle  between  two  JLines  in  Space. — The 

angle  0  between  two  right  lines  in  space  is  obviously 
equal  to  that  between  their  respective  parallels  through 
the  origin.  Hence,  by  formula  (A)  of  Art.  694, 

cos  6  =  cos  a  cos  a!  -\-  cos  /9  cos  ft'  +  cos  f  cos  f     (1) : 

which  expresses  the  angle  between  two  right  lines  in  terms 
of  their  direction- cosines. 


PLANE  ANGLES  IN  SPACE.  539 

Substituting  for  cos  «,  cos  a',  etc.,  from  the  preceding 
article,  we  get 


*  _  II'  -j-  mmf  -f-  nn'  ,y\  . 

~V(J2  +  m2  +  n2)  (I12  +  w'2  +  w'*) 


which  expresses  the  aftgrZe  between  two  right  lines  in  terms 
of  their  projections. 

Corollary  1,  —  The  condition  that  two  right  lines  in 
space  shall  be  parallel,  derived  from  (1),  is 

cos  a  cos  a'  -\-  cos  ft  cos  ft  -f  cos  7-  cos  f  —  1       (1), 

or,  derived  from  (2)  by  steps  analogous  to  those  in  the 
first  corollary  of  Art.  694, 

V          I  t     mf     _  m  .     n'     _  n  ,<^ 

^"=  m'     ~^'~~~~n'     J~~~~1 

Corollary  2.  —  The  condition  that  two  right  lines  in 
space  shall  be  perpendicular  to  each  other,  derived  from 
(1),  is 

cos  a  cos  a!  -f-  cos  ft  cos  ft'  -f  cos  f  cos  f  —  0       (1), 

or,  derived  from  (2), 

W  +  mm'  +  nn'  =  Q  (2). 

709.  Equation  to  a  Right  Line  perpendicular 
to  a  given  Plane.—  If  Ax  +  By  -f  Cz  +  i>  =  0  be  the 

given  plane,  the  required  equation  may  be  written 

x  —  a       y  —  b        z  —  c 
~A~        ~1T        ~~C~ 

For  we  may  suppose  the  perpendicular  to  pass  through 
any  fixed  point  abc;  and,  by  Art.  692,  its  direction- 
cosines  must  be  proportional  to  A,  B,  C. 

710.  Angle  contained  between  a  Right  Line 
and  a  Plane.  —  This  being  the  complement  of  the  angle 


540  ANALYTIC  GEOMETRY. 

contained  between  the   given  line  and  a  perpendicular 
to  the  plane,  if  the  given  line  be 

z  —  z' 


I 

we  have,  by  comparing  Arts.  708,  709, 
Al  +  Bm  +  On 


sin  6  = 


y  (A2  -f  B*  +  C2)  (F  -f-  m2  +  n?) 


Corollary, — The  condition  that  a  right  line  shall  be 
parallel  to  a  given  plane,  is 

Al  +  Bm  4-  On  =  0. 

711.  Condition  that  a  Right  Line  shall  lie 
wholly  in  a  given  Plane. — If  a  right  line  lies  wholly 
in  a  given  plane,  the  z  co-ordinate  resulting  from  an 
elimination  between  the  equation  to  the  plane  and  those 
of  the  line  must  of  course  be  indeterminate.  Hence,  if 
the  plane  be  Ax  -\-  By  -f  Cz  -j-  D  =  0,  and  the  line 
(x  =  mz  -\-  a,  y  =  nz  -j-  6),  so  that  we  have  by  elimi- 
nation A  (mz  -f  a)  -f  B  (nz  -f  b)  -f  Cz  +  D  =  0,  or 

Aa  +  M  +  D 
~  Am  +  En  +  0" 

we  must  have,  as  the  condition  required,  the  simultaneous 
relations 

Am  +  Bn+  0=  Aa  -f  56  +  Z>  =  0. 

Remark. — This  result  is  corroborated  by  the  fact,  that 
the  vanishing  of  the  numerator  of  z  indicates  that  the 
point  (a,  5,  0),  in  which  the  line  pierces  the  horizontal 
reference-plane,  is  in  the  given  plane ;  while  the  vanish- 
ing of  the  denominator  shows,  by  the  corollary  to  the 


EIGHT  LINES  MEETING  IN  SPACE. 


541 


previous  article,  that  the  line  is  parallel  to  the  given 
plane  :  two  conditions  which  obviously  place  the  line 
wholly  in  that  plane. 

712.  Condition  that  two  Right  Lines  in  Space 
shall  intersect.  —  Two  right  lines  in  space  will  not  in 
general  intersect,  because  the  four  equations 


x  =  mz  -+-  0, 
x  =  m'z  -+-  a', 


y  =  nz  -j- 
y  =  n'z  -j- 


being  in  general  independent,  are  not  compatible  with 
simultaneous  values  of  the  three  variables  #,  y,  z.  If, 
then,  the  two  lines  represented  by  these  four  equations 
do  intersect,  one  of  the  equations  must  be  derivable  from 
the  other  three,  and  the  condition  of  such  a  derivation 
will  be  the  required  condition  of  intersection. 

We  form  this  condition,  of  course,  by  eliminating  x,y,z 
from  the  four  equations.  To  do  this,  solve  the  first  and 
third,  and  also  the  second  and  fourth,  for  £,  and  equate 
the  two  values  thus  found.  The  result  is 


n  —  n' 


b  —  V 


EXAMPLES  INVOLVING  EQUATIONS  OF  THE  FIRST  DEGREE. 

1.  Show  that,  if  L,  M  and  N,  R  be  the  equations  to  two  inter- 
secting right  lines,  they  will  be  connected  by  some  identical  rela- 

tion 

IL  +  mM  +  nN  +  rR  =  0, 


and  that  the  plane  of  the  two  intersecting  lines  may  be  represented 
by  either  of  the  equations 

IL  +  mM^O,  nN+rR=--Q. 

2.  Find  the  equation  to  the  plane  which  passes  through  the 
lines 

x  —  a  _  y  —  b  _  z  —  c  x  —  a  _  y  —  b  _  z  —  c 

~~~       '~~~~  ~~~       ~             ~~~ 


542  ANALYTIC  GEOMETRY. 

3.  Find  the  equations  to  the  traces  of  any  given  plane  upon 
the  three  reference-planes,  and  prove  that  if  a  right  line  be  perpen- 
dicular to  a  given  plane,  its  projections  will  be  perpendicular  to  the 
traces  of  the  plane. 

4.  Find  the  equations  to  the  three  planes  which  pass  through 
the  traces  of  a  given  plane  upon  the  reference-planes,  and  are  each 
perpendicular  to  the  plane. 

5.  Find   the   equation    to   the   plane  which   passes  through   a 
given  right  line  and  makes  a  given  angle  with  a  given  plane. 

6.  If  (a',  /?',  /),   (a",  /?",  7X/)  be   the   direction-angles  of  two 
right  lines,  prove  that  the  direction-cosines  of  the  external  bisector 
of  the  angle  between  them,  are  proportional  to 

cos  of  •-(-  cos  a",       cos  /5'  +  cos  /?",       cos  /  -f  cos  /', 
and  that  those  of  the  internal  bisector  are  proportional  to 
cos  a'  —  cos  a",       cos  j8x  —  cos  /3X/,       cos  /  —  cos  y/x. 

7.  Three  planes  meet  in  one  point,  and  through  the  common 
section  of  each  pair  a  plane  is  drawn  perpendicular  to  the  third  • 
prove  that  in  general  the  planes  thus  drawn  pass  through  one  right 
line. 

8.  Find  the  equation  to  a  plane  parallel  to  two  given  right  lines, 
and  thence  determine  the  shortest  distance  between  the  lines. 

9.  A  plane  passes  through  the  origin:  find  the  bisector  of  the 
angle  between  its  traces  on  two  of  the  reference-planes. 

10.  Prove  that  the  locus  of  the  middle  points  of  all  right  lines 
parallel  to  a  given  plane,  and  terminated  by  two  fixed  right  lines 
which  do  not  intersect,  is  a  right  line. 


CHAPTER   THIRD. 

LOCUS  OF  THE  SECOND  ORDER  IN  SPACE.  | 

713.  The  general  equation  of  the  second  degree  in 
three  variables,  which  is  the  symbol  of  the  space-locus 
of  the  Second  order,  may  be  written 

Ax2  -f  ZHxy  -f  Bf  +  ZKyz  +  Ez*  +  ZLzx 

-f  2fe  +  2Fy  +  2Dz+C=Q      (1), 


SPACE-LOCUS  OF  THE  SECOND  ORDER.         543 

where  A,B,E;  H,K,L\   C,D,F,Gr  are  any  ten  con- 
stants whatever. 

Since  we  can  divide  this  equation  throughout  by  6Y,  it 
appears  that  the  number  of  independent  constants  is  nine. 
Hence,  nine  conditions  are  necessary  and  sufficient  to 
determine  the  locus.  Thus  we  learn  that,  for  example, 
the  space-locus  of  the  Second  order  is  a  surface  of  such 
a  form  that  one,  and  but  one,  such  surface  can  be  passed 
through  any  nine  points  which  do  not  lie  in  the  same  plane. 

714.  The  most  general  and  complete  criterion  of  the 
form  of  any  surface,  is  afforded  by  its  curves  of  section 
with  different  planes.  Let  us  apply  this  criterion  to  test 
the  figure  of  the  surface  denoted  by  (1). 

If  in  (1)  we  make  2  =  0,  that  is,  if  we  combine  (1) 
with  the  equation  to  the  horizontal  reference-plane,  we 
get 

Ax*  +  2Hxy  +  Eif  -f  2Gx  +  2Fy  +  0=  0, 

the  general  equation  to  the  Conic.  Hence,  as  we  can 
transform  the  reference-plane  of  XY  to  any  plane  that 
we  please,  and  as  such  a  transformation  will  not  affect 
the  degree  of  the  equation  of  section  just  found,  Every 
plane  section  of  a  surface  of  the  Second  order  is  a  conic. 
Moreover,  if  we  combine  (1)  with  z  =  /£,  that  is  (Art. 
695,  Cor.),  if  we  intersect  our  locus  by  any  plane  parallel 
to  the  plane  of  XY,  we  get 

Ax2  +  ZHxy  +  Bif  +  2G'x  +  ZF'y  +C'  =  0, 


where  Gr=G  +  kL,  Ff=F+kK,  C'  = 

Hence,  (Art.  669,)  since  the  co-efficients  A,H,B  remain 

unchanged  whatever  be  the  value  of  k,  The  sections  of  a 

surface  of  the  Second  order  by  parallel  planes  are  similar 

conies. 

An.  Ge.  49. 


544  ANALYTIC  GEOMETRY. 

To  denote,  then,  that  the  surface  of  the  Second  order 
is  represented  by  an  equation  of  the  second  degree  in 
space,  and  that  all  its  plane  sections  are  curves  of  the 
Second  order,  we  shall  henceforth  call  it  the  Quadric. 

THE  QUADBIC  IN  GENERAL. 

715.  To  increase  the  clearness  of  our  conception  of 
the  quadric  figure,  we  must  now  reduce  the  general 
equation  (1)  to  its  simplest  forms.  We  can  effect  this 
reduction  most  rapidly,  however,  by  taking  out  a  few 
leading  properties  of  the  surface,  partly  from  the  general 
equation  itself,  and  partly  from  the  results  of  its  first 
transformations. 

71O.  Let  us  transform  (1)  to  parallel  axes  through  a 
new  origin  x'y'z'.  Since  we  merely  have  to  write  x  -f-  x' 
for  x,  y  -f-  y'  for  y,  and  z  -f  2'  for  z,  it  is  easily  seen  that 
the  new  equation  will  be 

Ax2  +  Zffxy  -|-  Ef  -f  ZKijz  -f-  Ez2  -f  Lzx 

2D'z  -f  C'  =  0       (2), 


where  C'  ,  the  new  absolute  term,  is  the  result  of  substi- 
tuting x'y'z'  in  (1),  and  in  the  new  co-efficients  of  x,  y,  z 
we  have 

Gr  =  Ax'  -f-  Hy'  +  Lz'  -f  G, 
F'  =  Hxf  +  By'  +  Kz'  +  F, 
Df  ==  Lxr  -f  Ky'  +  Ez'  -f-  D, 

these  quantities  being  planar  functions  of  the  new  origin. 
It  deserves  especial  notice,  that  Gr  (Alg.,  411)  is  the 


#  For  the  discussion  of  Quadrics  in  complete  detail,  the  reader  is  referred 
to  SALMON'S  Geometry  of  Three  Dimensions,  from  which  the  investigations 
of  the  following  pages  have  in  the  main  been  reduced. 


DISCRIMINANT  OF  THE  QUADRIC.  545 

derived  polynomial  (or  derivative,  as  we  shall  call  it  for 
brevity)  of  C1  with  respect  to  x  ;  that  F1  is  the  derivative 
of  C'  with  respect  to  v/;  and  D1  ',  the  derivative  of  Cf  with 
respect  to  2.  Hence,  if  we  write  the  original  equation 
(1)  in  the  abbreviated  form  U  =  0,  we  may  use  for  the 
four  co-efficients  Cf  ,  G1  ',  jf7',  Dr  the  convenient  symbols 
Z7',  UJ,  U,',  U.'. 

TIT1.  As  we  shall  also  find  it  convenient  to  employ 
the  so-called  discriminant  of  the  equation  U=Q,  and 
several  of  its  derivatives,  we  will  determine  their  values 
before  advancing  farther. 

The  discriminant  of  any  function  may  be  defined  as 
the  result  obtained  by  solving  its  several  derivatives  for 
its  variables,  and  then  substituting  the  values  of  these 
in  the  function  itself.  Accordingly,  solving  for  .r,  y,  z  in 

Ux=^  Ax  +  Hy  +  Lz  +  G  =  Q, 
Uy  =  Hx  +  By  +  Kz  4-  F  =  0, 
Uz  =  Lx  +  Ky  -f  Ez  +  D  =  0, 

and  then  substituting  in  U,  we  get 

ABCE  +  2ADFK+  2BDGL  +  2CHKL  -f  2EFGH 


This,  then,  is  the  discriminant  of  the  given  quadric  U=  0, 
Snd  may  be  appropriately  represented  by  A. 

If  we  now  denote  the  several  derivatives  of  J,  taken 
with  reference  to  6r,  F,  D,  C  in  succession,  by  2#,  2/,  2c?,  c, 
we  shall  have 

g  =  BDL  -f  ^MT  —  5^(7  -h  GK2  —  DHK—  FKL, 
f  =  ADK+  EGH  —  AEF  +  FL2  •-  DHL  —  GKL, 
d  =  ^^7T+  5(y^  —  ABD  +  DH2  —  FHL  —  GHK, 
c  =  ^1^^  +  2fflK  —  ^^T2  —  BU  —  EH2. 


546  ANALYTIC  GEOMETRY. 


It  will  be  convenient  next  to  determine  the 
condition  upon  which  the  radius  vector  of  the  Quudric 
will  be  bisected  in  the  origin.  To  find  this,  throw  the 
general  equation  into  the  vectorial  form,  by  writing 
f)  cos  a  for  x,  p  cos  ft  for  y,  and  p  cos  f  for  2,  which  we 
may  evidently  do  if  a,  ft,  f  are  the  direction-angles  of 
the  radius  vector.  Equation  (1)  then  becomes 

{A  cos2  a  -f-  2-fiTcos  a  cos  /?  -f-  B  cos2  /? 

+  2-fiT  cos  p  cos  7  +  E  cos2  y-\-2L  cos  y  cos  a)  /J2 

+  2(£cosa-f  _Fcos/3+£>cos}')/>  +  (7=0. 


If  the  origin  bisects  the  radius  vector,  this  equation  will 
have  its  roots  numerically  equal  with  opposite  signs. 
Hence,  the  required  condition  of  bisection  is 

G  cos  a  -f  jPcos  ft  -f-  D  cos  f  =  0  ; 

or,  after  multiplying  through  by  />,  and  replacing  the 
corresponding  x9  y,  2,  we  learn  that  all  radii  vectores 
bisected  in  the  origin  must  lie  in  the  plane 


719.  If,  then,  in  the  equation  to  the  Quadric  we  had 
6r,  F,  D  all  =  0,  the  condition  of  bisection  would  be 
satisfied  for  all  possible  values  of  «,  ft,  ?  ;  or,  in  other 
words,  every  right  line  drawn  through  the  origin  to  meet 
the  quadric  would  be  bisected  in  the  origin,  and  the  origin 
would  be  a  center  of  the  quadric. 

720.  Resuming  now  our  transformations  of  equation 
(1),  let  us  suppose  the  new  origin  x'y'z'  to  which  (2)  is 
referred,  to  be  a  center.     The  new   6r,  F,  D  will  then 
vanish,  and  we  learn  (Art.  716)  that  the  center  lies  at 
the  intersection  of  the  three  planes 


QUADRICS  CENTRAL  AND  NON-CENTRAL.     547 

Solving  these  three  equations,  we  obtain,  as  the  co-ordi- 
nates of  the  center, 


x^  v  —  z  — 

—  ,      y  --  ,      z  ---  , 

c  c  c 

where  #,  /,  d,  c  have  the  values  given  in  the  table  of 
Art.  717. 

The  center,  then,  is  a  single  determinate  point,  and 
will  be  a  finite  real  one  if  c  is  not  zero,  but  not  other- 
wise. Hence,  Quadrics  are  either  central  or  non-central* 
and  central  quadrics  have  only  one  center. 

721.  By  taking,  then,  the  center  for  origin,  the  equa- 
tion to  any  central  quadric  may  be  written 


Ax2-\-  ZHxy  +%2-h  ZKyz  +Ez2+  2Lzx  +  C'=  0  (3), 

where  (Art.  716)  by  substituting  the  co-ordinates  of  the 
center  in  £7',  we  readily  find 

Gg  +  Ff  +  Dd  ±0c        A 

u  =  -  —  —  =  —  > 


e 


where  (r,  -JF,  D,  C  are  the  co-efficients  of  the  planar 
terms  of  (1),  and  #, /,  d,  <?,  J  have  the  meanings  assigned 
in  Art.  717. 


Let  us  next  inquire  into  the  form  of  the  diame- 
tral surfaces  of  the  Quadric. 

A  diametral  surface  of  a  given  surface  may  be  defined 
as  the  locus  of  the  middle  points  of  chords  drawn  par- 
allel to  a  given  right  line.  Suppose,  then,  that  «,  /9,  f 
are  the  direction-angles  common  to  a  system  of  chords 
in  a  quadric,  and  let  us  remove  the  origin  of  equation 
(1)  to  any  point  on  the  locus  of  the  middle  points  of  the 


548  ANALYTIC  GEOMETRY. 

system.     By  Art.  718,  the  new  co-efficients  6r',  Fr,  Df 
must  then  fulfill  the  condition 

Gf  cos  a  +  F'  cos  /3  +  D'  cos  f  =  0, 

and  the  equation  to  the  diametral  surface  of  the  Quadric 
will  therefore  be  (Art.  716) 

Ux  cos  «  -f-  Uv  cos  /3  -f  Uz  cos  7-  =  0. 

This  (Art.  698,  Cor.)  denotes  a  plane  passing  through 
the  intersection  of  the  three  planes  Ux,  Uy,  Uz,  namely 
(Art.  720),  through  the  center;  arid,  as  the  direction- 
angles  «,  /9,  Y  are  arbitrary,  we  have  the  theorem : 
Every  surface  diametral  to  a  quadric  is  a  plane  passing 
through  the  center,  and  every  plane  passing  through  the 
center  of  a  quadric  is  a  diametral  plane. 

It  should  be  observed,  of  course,  that  this  theorem 
applies  to  the  non-central  quadrics  only  by  regarding  a 
point  infinitely  distant  from  the  origin  as  their  center. 
Such,  indeed,  is  the  fact  indicated  by  the  central  co- 
ordinates (Art.  720)  g  :  c,  f :  c,  d  :  <?,  in  which  c  =  0  for 
the  non-central  quadrics.  But  if  the  diametral  planes 
pass  through  a  common  point  at  infinity,  their  several 
common  sections  will  meet  in  a  point  at  infinity;  in 
other  words,  will  be  parallel.  .  Hence,  The  diametral 
planes  of  a  non-central*  *****  Lare  parallel  to  a  fixed 
right  line. 

723.  The  diametral  planes  which  bisect  chords 
parallel  to  the  axis  of  #,  the  axis  of  ?/,  and  the  axis 
of  z  respectively,  are  found  by  successively  supposing 
ft  =  r  =  90°,  r  =  a  =  90°,  a  =  p  =  90°  in  the  equation 
of  the  preceding  article.  They  are  therefore  respectively 


CONJUGATE  PLANES  AND  DIAMETERS.        549 

UK  —  0,    Uy=Q,   Uz  =  0  ;   or,  writing  the  abbreviations 
in  full, 

AX  +  Hy  +-LZ  +  a  =  o, 

+  %  +  Kz  -f  .F  =  0, 

-f  .#z  -f  D  =  0. 


For  brevity,  a  diametral  plane  is  said  to  be  conjugate 
to  the  direction  of  the  chords  which  it  bisects.  Now  the 
condition  that  the  plane  Ux,  which  is  conjugate  to  the 
axis  of  .r,  may  be  parallel  to  the  axis  of  y,  according  to 
the  corollary  of  Art.  696  is  H  =  0.  But,  obviously, 
this  is  also  the  condition  that  the  plane  Uy,  which  is 
conjugate  to  the  axis  of  y,  may  be  parallel  to  the  axis 
of  x.  Hence,  as  the  co-ordinate  axes  may  have  any 
direction,  If  a  diametral  plane  conjugate  to  a  given 
direction  be  parallel  to  a  given  right  line,  the  plane  con- 
jugate to  this  line  will  be  parallel  to  the  first  direction. 

724.  If  in  our  general  equation  (1)  we  had  H,  jfiT,  L 
all  ==  0,  the  equations  of  the  preceding  article  would  be 
reduced  to 


The  diametral  planes  conjugate  to  the  three  axes  would 
thus  (Art.  695,  Cor.)  become  parallel  to  the  reference- 
planes;  and,  by  the  theorem  'last  proved,  each  would  be 
conjugate  to  the  common  section  of  the  other  two. 

Three  diametral  planes  thus  related  are  culled  conju- 
gate planes,  and  the  three  right  lines  in  which  they  cut 
each  other  two  and  two  are  called  conjugate  diameters. 
Three  diameters  are  therefore  conjugate,  when  each  is 
conjugate  to  the  plane  of  the  other  two. 

We  thus  reach  the  important  result,  that  whenever  the 


550  ANALYTIC  GEOMETRY. 

equation  to  a  quadric  lacks  the  co-efficients  H,  K,  L,  the 
co-ordinate  axes  to  which  it  is  referred  are  parallel  to  a 
set  of  conjugate  diameters  ;  and,  conversely,  that  by  em- 
ploying axes  parallel  to  a  set  of  conjugates,  we  can  always 
cause  these  co-efficients  to  vanish  from  the  equation  to  a 
central  quadric. 


As  the  foregoing  argument  evidently  does  not 
conflict  with  the  supposition  all  along  made,  that  the  co- 
ordinate axes  are  rectangular,  it  follows  that  every  central 
quadric  has  one  set  of  conjugate  planes  and  diameters  which 
are  at  right  angles  to  each  other. 

In  fact,  diametral  planes  perpendicular  to  the  chords 
which  they  bisect,  or  principal  planes,  as  they  are  called, 
exist  in  all  quadrics  whether  central  or  not  ;  though  the 
triconjugate  groups,  of  course,  are  peculiar  to  central 
quadrics.  For  if  we  seek  the  condition  that  the  plane 
(Art.  722) 

Ux  cos  a  -f-  Uu  cos  /9  -f  Uz  cos  f  =  -  0 

may  be  perpendicular  to  its  conjugate  chords,  the  prin- 
ciple (Art.  709)  that  the  direction-cosines  of  the  chords 
must  be  proportional  to  the  co-efficients  of  the  plane, 
gives  us,  if  we  put  k  =  the  constant  ratio  between  these 
quantities, 

A  cos  «  -f-  H  cos  /9  -f-  L  cos  ?  =  k  cos  «, 
II  cos  a  -f-  B  cos  /?  -(-  ./Tecs  7-  =  k  cos  /?, 
L  cos  a  -f-  K  cos  /9  -f-  E  cos  f  •=  k  cos  j. 

Eliminating  cos  «,  cos  /9,  cos  7-  from  these  equations,  the 
required  condition  is 


THE  PRINCIPAL  PLANES.  551 


where  c  has  the  same  value  as  in  Art.  717.  Having 
thus  a  cubic  for  determining  the  ratio  7c,  we  learn  that 
a  quadric  has  in  c/cncral  three,  and  only  three,  principal 
planes. 

In  the  non-central  quadrics  however,  since  in  them 
(Art.  721)  we  have  c  =  0,  one  of  the  roots  of  this  cubic 
must  be  0,  and  the  equation  to  one  of  the  principal  planes 
will  therefore  assume  the  form 

Ox  +  0#  +  02  +  constant  =  0. 

In  the  non-central  quadrics,  therefore,  by  the  analogy 
of  Art.  110,  the  third  principal  plane  is  situated  at 
infinity. 

726.  We  are  now  prepared  to  put  our  general  equa- 
tion into  its  simplest  forms. 

First,  let  us  suppose  that  the  derivative  c  is  not  zero. 
From  equation  (3)  in  Art.  721,  which  is  already  referred 
to  the  center  as  origin,  we  can  at  once  proceed  by  taking 
for  new  axes  a  set  of  conjugate  diameters;  and,  as  we 
still  adhere  to  rectangular  co-ordinates,  let  these  new 
axes  be  conjugate  to  the  principal  planes.  Then  (Art. 
724)  the  co-efficients  IT,  K,  L  vanish  from  (3),  and  the 
equation  to  any  central  quadric  takes  the  form 

4/ar5  +  #y+^/32+Cr'  =  0  (4). 

Secondly,  suppose  that  c  is  equal  to  zero.  We  can 
not  then  arrive  at  the  form  (3)  ;  but,  going  back  to  (1), 
we  may  first  change  the  direction  of  the  rectangular 
axes,  and  then  remove  the  origin.  Now  it  can  readily 


552  ANALYTIC  GEOMETRY. 

be  shown,  that,  in  passing  from  one  set  of  rectangular 
axes  to  another,  the  new  co-efficients  of  x2,  ?/2,  z2  (or  A', 
Bf,  E')  are  the  three  roots  of  the  cubic  * 

u?  —  (A  +  B  -f  E)  u2 


*  We  append  Salmon's    proof  of  this,  as  it  is  remarkable  for  its  brev- 
ity.    See  his  Geometry  of  Three  Dimensions,  p.  50. 

"  Let  us  suppose  that  by  using  the  most  general  transformation,  which 
is  of  the  form 


x  —  \x  +  py  +  vz,      y=\'x  +  p'y  +  v'Zf       z  =  \"x  +  p"y~+ 
that  Ax*  +  2Hxy  +  By*  +  2Kyz  +  Ez*  +  2Lzx 

becomes  A'~x*  +  2H'~xy  +  B'~y*  +  2K'~i/z  +  E'~z*  +  2L'~zx~, 


which  we  write  for  shortness  U=U.     If  both  systems  of  co-ordinates  be 
rectangular,  we  must  have 


which  we  write  for  shortness  S=  S.  Then  if  k  be  any  constant,  we  must 
have  U+kS=  U+  kti.  And  if  the  first  side  be  resolvable  into  factors, 
so  must  also  the  second.  The  discriminants  of  U -\-  kS  and  U  +  kti  must 
therefore  vanish  for  the  same  values  of  k.  But  the  first  discriminant  is 

A3  -  (A  +  B  +  E )  k*  +  (AB  +  BE  +  EA  -  W  -  K'2  -  L1}  k  -  c. 

Equating  then  the  co-efficients  of  the  different  powers  of  k  to  the  corre- 
sponding co-efficients  in  the  second,  we  learn  that  if  the  equation  be 
transformed  from  one  set  of  rectangular  axes  to  another,  we  must  have 

A  +  B+  E=  A'  +  B'  +  E', 
AB  +  BE  +  EA  -  m  -  K*-  V  =  A' B'  +  B'E'+  E' A'  -  H'*-K'*-L">, 

ABE  +  2HKL  -  A  K 2  -  BL'2  -  Elf2  = 

A'B'E'  +  IH'K'L'  -  A'K't-B'L't-E'H''2." 

By  solving  these  three  equations  for  cither  A',  B' ,  or  E',  we  obtain 
the  cubic  in  the  text  above,  where  u  is  merely  a  symbol  for  the  unknown 
co -efficient. 


THE  QUADRIGA  CLASSIFIED.  553 

Hence,  as  we  now  have  c  —  0,  one  of  the  roots  of  this 
cubic,  and  therefore  one  of  the  new  co-efficients  A', 
B',  E',  must  vanish  whatever  be  the  directions  of  the 
new  rectangular  axes.  By  taking  these  axes  parallel  to 
the  principal  planes,  thus  causing  the  new  H1  ',  K',  L'  to 
disappear,  we  can  therefore  reduce  the  original  equation 
to  the  form 


B'f  -f  E'#  +  2G'x  +  2F'y  +  Wz  +  C=  0, 

as  this  transformation  (Art.  685)  does  not  affect  the 
absolute  term.  And  now  we  can  remove  the  origin  to 
the  point  in  which  the  line  Fr  =  0,  Df  =  0  pierces  the 
Quadric,  thus  destroying  the  absolute  term  as  well  as 
the  co-efficients  of  y  and  z.  The  equation  to  a  non- 
central  quadric  will  then  take  the  form 


B'y*  + 
or,  as  it  may  be  more  symmetrically  written, 

if  +  Qz2  =  Px  (5). 

Equations  (4)  and  (5)  are  the  simplest  forms  of  the 
space-equation  of  the  second  degree. 

CLASSIFICATION  OF  QUADRICS. 

T2T.  By  means  of  equations  (4)  and  (5),  we  can  now 
ascertain  the  several  varieties  of  quadric  surfaces,  and 
the  peculiar  figure  of  each. 

728.  We  begin  with  the  CENTRAL  QUADRICS,  repre- 
sented by  the  equation 

A'x2  -f  B'if  -f  E'z2  +  C1  =  0. 


554  ANALYTIC  GEOMETRY. 

I.  Let  A',  B',  E'  all  be  positive.     Then,  if  C"  is  neg- 
ative, the  equation  can  at  once  be  put  into  the  form 


where  a,  6,  c  are  the  lengths  of  the  intercepts  cut  off 
upon  the  axes  of  x,  y,  and  z  respectively.  The  sections 
of  the  surface  with  any  planes  of  the  form  z  =  &,  x  =  Z, 
y  =  m,  are  the  ellipses 

^     t^_       p     f     z^       _P_    i2     *2_-,_^ 
a*~*~b*~          c2  '    6*"1V~          a2'    e*~ra5i  62 

These  are  real  for  every  positive  or  negative  value  of  k, 
I,  m  that  is  not  greater  respectively  than  a,  6,  c',  but  are 
imaginary  for  all  greater  values.  The  quadric,  therefore, 
lies  wholly  inside  the  rectangular  parallelepiped  formed 
by  the  six  planes  z  =  ±:  c,  x  =  ±  a,  y  =  ±b,  but  is 
continuous  within  those  limits,  and,  having  elliptic  sec- 
tions with  the  reference-planes  and  all  planes  parallel 
to  them,  is  properly  called  the  Ellipsoid. 

Its  semi-axes  are  of  course  respectively  equal  to  a,  b, 
c,  and  we  suppose  that  the  reference-planes  are  so  taken 
that  a  is  in  general  greater  than  ft,  and  b  greater  than 
c.  But  the  following  particular  cases  must  be  considered: 

1.  If  b  =  c,  the  section  with  any  plane  x  —  I  becomes 
y2  -j-  z2  =  constant,  and  is  therefore  a  circle.     The  surface 
may  then  be  generated  by  revolving  an  ellipse  upon   its 
axis  major,  and  is  called  an  ellipsoid  of  major  revolution; 
or,  with  greater  exactness,  the  Prolate  Spheroid. 

2.  If  b  =  a,  the  section  with  any  plane  z  =  Jc  becomes 
a  circle,  the  surface  may  be  generated  by  revolving  an 
ellipse  upon  its  minor  axis,  and  is  therefore  called  an 
ellipsoid  of  minor  revolution  ;  or,  the  Oblate  Spheroid. 


THE  QUADEICS  CLASSIFIED.  555 

3.  If  a  =  b  =  c  =  r,  all  the  plane  sections  of  the  surface 
are  circles,  and  the  equation  becomes 


which  is  therefore  the  equation  to  the  Sphere. 

Next,  if  0'  is  zero,  our  general  central  equation,  taking 
the  form 

'z2  =  0, 


can  only  be  satisfied  by  the  simultaneous  values 

x  =  0,       y  =  0,       z  =  0, 

and  thus  denotes  the  Point,  which  may  therefore  be 
regarded  as  an  infinitely  small  ellipsoid. 

Finally,  if  Cf  is  positive,  the  general  equation  becomes 

x2        i/2        z1 

a*  +  F  +  ?  =     71' 

which  having  the  ellipsoidal  form  in  its  first  member, 
but  involving  an  impossible  relation,  may  be  said  to 
denote  an  imaginary  ellipsoid. 

II.  Let  A'  and  B'  be  positive,  but  E'  be  negative. 
Then,  if  C'  be  also  negative,  we  can  write  the  equation, 
in  terms  of  the  intercepts, 


+     _    -  1 

a2  1    62        c2 


Here,  the  sections  formed  by  the  planes  x  =  I,  y  = 
are  the  hyperbolas 

f        z2  __          I2  3?        ^  —\       ^L 

~~~=        ~9  ~~~'~         ~ 


556 


ANALYTIC  GEOMETRY. 


which  are  real  for  all  values  of  I  and  ???,  but  whose  branches 
cease  to  lie  on  the  right  and  left  of  the  center,  and  are 
found  above  and  below  it,  when  I  >  a  and  m  >  b.  The 
section  by  any  plane  z  =  k  is  an  ellipse 


5>f- 

a"        er 


-1-  -3- 


which  being  real  for  every  value  of  A",  the  surface  is 
continuous  to  infinity.  This  being  so,  and  the  sections 
by  the  vertical  reference-planes  and  all  their  parallels 
being  hyperbolas,  the  surface  is  called  the  Hyperboloid 
of  One  Nappe. 

The  quantities  a,  6,  c  are  called  its  semi-axes,  though 
it  is  evident,  by  making  x  =  y  =  0  in  the  equation,  that 
the  axis  of  z  does  not  meet  the  surface.  The  real  mean- 
ing of  c  will  soon  appear.  In  general,  a  is  supposed 
greater  than  b.  In  case  we  have  a  =  b,  the  sections 
parallel  to  the  plane  XY  become  circles,  the  surface 
may  be  generated  by  revolving  an  hyperbola  upon  its 
conjugate  axis,  and  is  called  an  hyperboloid  of  revolution 
of  one  nappe. 

Next,  when  C'  =  0,  the  equation  assumes  the  form 


The  section  made  by  the  reference-plane  z  =  0,  is  the 
point  ^L'o;2  -f-  -S'?/2  — -  0,  while  that  by  any  parallel  plane 
z  =  k,  is  the  ellipse  A'x2  -f-  By2  =  E'k2.  The  sections 
formed  by  the  planes  x  =  0,  y  =  0,  are  pairs  of  inter- 
secting right  lines,  B'y1 —  JEfz'2  =  Q  and  A'x2  —  E'z1  =  ^\ 
as  also  the  section  by  any  vertical  plane  y  =  mx,  is  the 
pair  of  lines  (A'  -f  m2B'}  x2  —  E'z2=  0.  The  surface  is 
therefore  a  Cone,  whose  vertex  is  the  origin.  If  we  have 

j  *  -I-  T\  /  J-  T~r  /  J- 


THE  QUAD  RIGS  CLASSIFIED.  557 

the  cone  is  said  to  be  asymptotic  to  the  preceding  hyper- 
boloid.  If  a  =  6,  or  A'  =  B',  the  section  by  the  plane 
z  =  k  is  the  circle  x2  -}-  y2  —  constant,  and  the  cone  is  a 
circular  one. 

Finally,  when  Cf  is  positive,  by  changing  the  signs 
throughout  we  may  write  the  equation 

z-  __  —    -y—  =  \ 
c2        a2        b2 

The  plane  z  =  k  now  evidently  cuts  the  surface  in  imag- 
inary ellipses  so  long  as  k  <  c,  but  in  real  ones  when  k 
passes  the  limit  c  whether  positively  or  negatively.  The 
surface  therefore  consists  of  two  portions,  separated  by 
a  distance  =  20,  and  extending  to  infinity  in  opposite 
directions.  The  sections  by  the  planes  x  =  l,  y  =  m, 
are  hyperbolas.  The  surface  is  therefore  called  the 
Hyperboloid  of  Two  Nappes. 

By  making  x  =  y  =  0,  we  find  that  the  intercept  of 
this  surface  on  the  axis  of  z  is  —  c  ;  while  the  intercepts 
upon  the  axes  of  x  and  y,  found  by  putting  y  =  z  =  Q  and 
z  =  x  =  Q,  are  the  imaginary  quantities  aV  —  1,  hi/  —  1. 
Moreover,  the  sections  by  the  planes  #  —  0,  #  —  0,  being 

^     1JL  —  i    !i     ?£  —  i 

2~~    2~      '       2  ~~    2~ 


are  hyperbolas  Conjugate  to  those  in  which  the  same 
planes  cut  the  Hyperboloid  of  One  Nappe.  We  there- 
fore perceive  that  the  present  hyperboloid  is  conjugate 
to  the  former,  and  that  the  real  meaning  of  c  in  the 
equation  to  the  former  is,  the  semi-axis  of  its  conjugate 
surface. 

When  a  =  b,  this  hyperboloid  also  becomes  one  of 
revolution,  and  is  called  an  hyperboloid  of  revolution  of 
two  nappes. 


558  ANALYTIC  GEOMETRY. 

III.  Let  A'  be  positive^  and  B'  and  E'  both  negative. 
The  general  equation  may  then  be  written 

^        f_        ^  __  -i 

•'  I'  v>    ---        * 

a"        or        eH 

when  (7'  is  negative  ;  or 


__  _ 

b*  "    c2  ""  a2"  - 

when  C'  is  equal  to  zero  ;  or 

y"  L  -     -  -  i 

fr-'   "     c2  ""  «2~ 

when  Cr  is  positive.  The  present  hypothesis  therefore 
presents  no  new  forms,  but  merely  puts  those  of  the  pre- 
ceding supposition  into  a  different  order.  It  is  usual, 
however,  to  write  the  equation  to  the  Hyperboloid  of  Two 
Nappes  in  the  form 

tf        £        z2 
a2        b*  '     c*~ 

rather  than  in  that  obtained  by  the  final  supposition  of  II. 
The  advantage  of  doing  so  appears  in  connection  with  the 
hyperboloids  of  revolution  ;  for,  if  we  make  b  =  c  in  the 
above  equation,  we  learn  that  an  hyperboloid  of  revolution 
of  two  nappes  may  be  generated  by  revolving  an  hyper- 
bola upon  its  transverse  axis.  Under  the  present  hypoth- 
esis, therefore,  the  hyperboloids  of  two  nappes  and  of  one 
may  be  generated  by  revolving  the  same  hyperbola,  first 
upon  its  transverse,  and  then  upon  its  conjugate  axis. 
Under  the  hypothesis  of  II,  on  the  contrary,  the  two 
surfaces  would  be  generated  by  a  pair  of  conjugate 
hyperbolas  revolving  upon  the  same  axis. 


THE  QUADRICS  CLASSIFIED.  559 

«  Let    us,    secondly,    consider    the   NON-CENTRAL 
QUADRICS,  represented  by  the  equation 

if  +  Qz2  =  Px. 

I.  Suppose  Q  to  be  positive.  Any  plane  x  ==  I  cuts 
the  surface  in  an  ellipse  y2  +  Qz2  =  Ply  which,  if  P  is 
positive,  will  be  real  only  on  condition  that  I  is  not 
negative;  or,  if  P  is  negative,  only  on  condition  that  I 
is  not  positive.  The  surface,  then,  consists  of  a  single 
shell,  extending  to  infinity  on  one  side  of  the  plane  YZ, 
and  in  fact  touched  by  that  plane  in  the  point  y2-\-Qz2  =  Q, 
that  is,  in  the  origin.  The  sections  of  this  shell  by  the 
planes  y  =  0,  2  =  0,  are  the  parabolas  z2  =  (P  :  Q)  x, 
y*  —  Px.  Hence,  all  sections  by  planes  parallel  to  these 
are  also  parabolas,  and  the  surface  is  the  Elliptic  Para- 
boloid. 

1.  If  Q  =  1,  the  section  by  the  plane  x  =  I  is  the  circle 
y2Jrz2  =  Pl^  and  the  surface  may  be  generated  by  revolv- 
ing a  parabola  upon  its  principal  axis.     It  is  then  called 
a  paraboloid  of  revolution. 

2.  We  have  thus   far   not  made   P  =  0,  because  the 
transformation  to  y2  +  Qz2  =  Px  in  general  excludes  that 
supposition,  being  made  upon   the  assumption   that   we 
can  not,  in  the  equation  (Art.  726) 


+  E'z2  +  2G'x  +  ZF'y  +  2D'z  +(7=0, 

destroy  all  the  three  co-efficients  Gf,  F',  D'  together. 
But  if  Gr  were  itself  =  0,  then  any  point  on  the  line 
F'  =  0,  D'  =  0  would  be  a  center  of  the  quadric  ;  and 
as  this  line  would  thus  pierce  the  surface  only  at  infinity, 
we  could  not,  by  placing  the  origin  at  the  piercing-point, 
cause  the  absolute  term  to  disappear.  However,  by  taking 
An.  Ge.  50. 


560  ANALYTIC   GEOMETRY. 

the  origin  upon  the  line  F'  =  0,  Dr  =  0,  the  equation 
would  be  reduced  to  the  form 

f  +  Qz*  =  R. 

This  equation,  at  first  sight,  appears  to  represent  an 
ellipse  in  the  plane  YZ.  But,  obviously,  it  is  true  not 
only  for  points  whose  x  =  0,  but  for  points  answering  to 
any  value  of  x  that  corresponds  to  a  given  y  and  z. 
It  denotes,  then,  a  cylinder  whose  base  is  the  ellipse 
y2  +  Qz2  —  R)  an(i  whose  axis  is  the  axis  of  x.  Hence 
we  learn  that  a  particular  case  of  an  elliptic  paraboloid 
is  the  Elliptic  Cylinder. 

When  R  =  0,  this  cylinder  breaks  up  into  two  imag- 
inary planes,  whose  common  section,  however,  being 
projected  in  the  real  point  y2  -j-  Qz2  —  0,  is  a  real  right 
line,  perpendicular  to  the  plane  YZ. 

II.  Suppose  Q  =  Q.  The  equation  ?/2+  Qz2  =  Px  then 
becomes 


and  therefore  denotes  the  Parabolic  Cylinder. 

By  shifting  the  origin  along  the  axis  of  #,  the  equation 
to  this  cylinder  takes  the  more  general  form 

f  =  Px  +  N. 

When,  therefore,  P=  0,  this  cylinder  breaks  up  into  the 
two  parallel  planes 


which  are  real,  coincident,  or  imaginary,  according  as 
N  is  positive,  equal  to  zero,  or  negative. 

III.  Suppose  Q  to  be  negative.     Then  the  surface 


ANALOGIES  OF  QUADRICS  TO  CONICS.         561 

will  meet  all  planes  parallel  to  y  =  0  and  z  =  0  in  par- 
abolas ;  but,  being  met  by  any  plane  x  =  l  in  the  real 
hyperbola  if  +  Qz*  =  PI,  is  called  the  Hyperbolic  Par- 
aboloid. 

It  evidently  meets  the  plane  x  =  0  in  the  two  inter- 
secting lines  y~-\-  Qz2  =  Q,  and  extends  to  infinity  on  both 
sides  of  that  plane.  As  a  particular  case, 

1.  Corresponding  to  the  negative  Q,  we  have  the 
cylinder 


As  this  meets  the  plane  x  =  0  in  the  hyperbola  y1  -f-  Qz2 
=  R,  it  is  called  the  Hyperbolic  Cylinder. 

When  R  =  0,  this  evidently  breaks  up  into  two  inter- 
secting planes. 

*73O.  The  foregoing  include  all  the  varieties  of  the 
Quadric.  By  means  of  the  several  equations  contained 
in  the  two  preceding  articles,  we  could  now  proceed  to 
develop  all  the  known  properties  of  these  surfaces.  The 
student,  however,  can  hardly  have  failed  to  observe  the 
remarkable  analogy,  not  only  in  respect  to  the  preceding 
classification  and  its  corresponding  equations  but  in  re- 
spect to  such  properties  as  have  already  been  developed, 
which  subsists  between  these  surfaces  and  the  several 
varieties  of  the  Conic.  He  will  therefore  anticipate  that 
by  applying  to  the  equations  we  have  just  obtained,  the 
methods  with  which  the  discussion  of  conies  has  now 
thoroughly  familiarized  him,  he  can  obtain  the  analogous 
properties  of  quadrics  for  himself.  Accordingly,  several 
of  the  more  important  ones  have  been  presented  in  the 
examples  at  the  close  of  this  Chapter. 

It  may  deserve  mention,  however,  that  in  these  quadric 
analogies  to  conies,  lines  generally  take  the  place  of 


562  ANALYTIC  GEOMETRY. 

points,  and  surfaces  the  place  of  lines ;  also,  that  where 
conic  elements  go  by  twos,  the  quadric  elements  gene- 
rally go  by  threes.  Thus,  the  conception  of  two  conju- 
gate diameters  is  replaced  by  that  of  three  conjugate 
planes. 

Often,  in  connection  with  this  principle  of  substitution, 
the  result  of  the  analogy  is  altogether  unexpected.  For 
instance,  the  quadric  analogue  of  the  conic  focus,  is  itself 
a  conic,  known  as  the  focal  conic,  which  lies  in  either 
of  the  principal  planes  ;  so  that,  in  general,  every  quadric 
has  three  infinitudes  of  foci. 

In  fact,  the  subject  of  Foci  and  Confocal  Surfaces  is 
one  of  the  most  recent  as  well  as  the  most  intricate  in 
connection  with  quadrics.  It  was  originally  investigated 
by  CHASLES  and  MACCULLAGH  independently ;  of  whose 
discoveries  Salmon  has  given  a  brilliant  account  in  the 
Abridged  Notation.* 

SURFACES  OF  REVOLUTION  OF  THE  SECOND  ORDER. 

731.  Any  surface  that  can  be  generated  by  revolving 
any  curve  about  a  fixed  right  line  is  called  a  surface  of 
revolution.  The  revolving  curve  is  named  the  generatrix, 
and  the  fixed  line  around  which  it  moves  is  termed  the 
axis. 

We  came  upon  the  Quadrics  of  Revolution  and  their 
equations,  in  the  preceding  investigations.  But  we  shall 
here  give  some  account  of  them  from  another  point  of 
view,  for  the  sake  of  putting  the  student  in  possession 
of  the  general  method  of  revolutions. 

Let  the  shaded  surface  in  the  diagram  represent 


*See  his  Geometry  of  Three  Dimensions,  p.  101. 


SURFACES  OF  REVOLUTION. 


563 


the  surface  generated  by  any  curve,  revolving  about  an 
axis  00.     Let  the  equa- 
tions to  the  "projecting  z/1 

cylinders"  of  the  gener- 
atrix be 

«=./»»   y=<p(*)- 

Then,  from  the  definition 

of  a  surface  of  revolution, 

the   co-ordinates   of  any 

point   P  on   the   surface 

must  at  once  satisfy  the 

conditions  for  being  in  the 

generatrix   and  in  a   circle   perpendicular  to   the    axis. 

Hence,  if  r  =  the  distance  PR  of  any  point   on  the 

surface  from  the  axis  Z'Z,  we  shall  have  simultaneously 


Eliminating  the  indeterminate  r  from  these   equations, 
we  get  2  2 


which  expresses  the  uniform  relation  among  the  variable 
co-ordinates  of  the  surface,  and  is  therefore  the  general 
equation  to  any  surface  of  revolution. 

This  becomes  the  equation  to  the  surface  generated  by 
a  given  curve,  when  we  expand  f  (z)  and  <p  (z)  in  accord- 
ance with  the  equations  to  the  generatrix.  Were  we  to 
suppose  the  generatrix  in  one  of  the  reference-planes, 
say  the  plane  ZX,  which  we  may  always  do  when  the 
generatrix  intersects  its  axis,  we  should  have  y  —  tp  (z)  =  0, 
and  the  general  equation  would  assume  the  simpler  form 


564  ANALYTIC  GEOMETRY. 

In  this,  we  suppose  the  axis  of  z  to  be  the  axis  of 
revolution ;  but,  obviously,  analogous  equations  of  revo- 
lution about  the  axis  of  x  or  of  y  are 


733.  Equation  to  the  Right  Circular  Cone.  —  Let 

the  axis  of  the  cone  be  the  axis  of  2,  and  its  base  the 
plane  XY.  Then,  if  the  co-ordinates  of  the  vertex  be 
x'  =  0,  zr  =  c,  the  equation  to  the  generatrix  (Art.  101, 
Cor.  1)  will  be  x  =  m(z  —  c),  where  (Art.  705)  w  —  coty?, 
if  <p  =  the  inclination  of  the  side  to  the  base  of  the  cone. 
Hence 

.  ,         z  —  c 


and,  substituting  in  the  equation  of  revolution,  we  get 

(x2  -j-  y2)  tan2  <p  =  (z  —  c)2 
as  the  required  equation  to  a  right  circular  cone. 

734.  Section  of  a  Right  Circular  Cone  by  any 
Plane.  —  Since  the   sections   formed  by  parallel  planes 
are  similar  (Art.  714),  it  will  be 
sufficient  to  consider  the  section 
formed  by  any  plane  NBL  passing 
through  the  axis   of  y.     As  this 
plane  is  projected  upon  the  plane 
ZX  in  the  line  OL,  its  equation 
may  be  written 

z  —  x  tan  0, 

where   x  =  OQ,  the   abscissa   of 

any  point  P  in  the  plane  ;  z  =  QM,  the  corresponding 

ordinate  ;  and  6  —  the  angle  A'OL,  which  measures  the 


CONE  AND  ITS  SECTIONS.  565 

inclination  of  the  plane  to  the  plane  XY.  Substituting 
for  z  in  the  above  equation  to  the  cone,  we  obtain 

(x2  +  f)  tan2  <f>  =  (x  tan  0  —  c)2, 

the  equation  to  the  curve  of  section  NBL.  Transform- 
ing this  to  its  own  plane,  we  shall  have,  in  the  formulas 
of  Art.  685,  a  =  6,  ft  =  90° ;  a!  =  90°,  /?'  =  0  ;  a"  =  90°, 
ft"  =  90°.  We  therefore  replace  x  by  x  cos  0,  and  leave 
y  unchanged,  thus  obtaining,  as  the  equation  to  any 
plane  section  of  a  right  circular  cone, 

x2  (tan2^>  —  tan2  0)  cos2  0  +  ?/2tan2  <p  -f  2cx  sin  6  =  c\ 

in  which  <p  =  the  angle  OAC,  and  6  =  the  angle  A'OL. 

735.  The  Curves  of  the  Second  Order  are  Con- 
ies.— The  proof  of  this  theorem,  promised  in  Art.  633, 
is  furnished  by  the  equation  just  obtained.  For  this 
evidently  conforms  to  the  type 

Ax2  -f  ZHxy  +  By2  +  2Gx+2Fy  +  C  =  0, 
giving  to  the  general  co-efficients  the  particular  values 

A  =  (tan2  <p  —  tan2  0)  cos2  0,     H=Q,     B  =  tan2  <p, 
G  =  csin0,  F=Q,     C=  —  c2. 

It  therefore  denotes  a  curve  of  the  Second  order,  whose 
species,  depending  on  the  sign  of  H2  —  AS,  in  fact  here 
depends  on  the  sign  of  A ;  since  _ff—  0,  and  B  is  neces- 
sarily positive. 

I.  Let  A  be  positive.  The  function  H2  —  AB  will 
then  be  negative,  and  (Art.  158)  the  section  will  be  an 
ellipse.  But  if  A  is  positive,  tan2  0  must  be  less  than 
tan2  <p  ;  or,  we  shall  have 


566  ANALYTIC  GEOMETRY. 

That  is,  If  the  inclination  of  the  secant  plane  be  less  than 
that  of  the  side  of  the  cone,  the  section  will  be  an  ELLIPSE. 
1.  One  form  of  the  condition  6  <C  <p  is  6  ==  0.  But 
under  this  supposition,  the  equation  to  the  section 
assumes  the  form 


and  we  obtain  a  circle,  as  a  particular  case  of  the  Ellipse. 
2.  If  under  the  condition  6  <  <p,  we  suppose  c  =  0, 
or  that  the  secant  plane  passes  through  the  vertex  of 
the  cone,  the  section  becomes 

x2  (tan2  <p  —  tan2  0)  cos2  6  +  f  tan2  <p  =  0, 
and  we  have  a  point,  as  the  limiting  case  of  the  Ellipse. 

II.  Let  A  =  0.  The  function  H2  —  AB  will  then  also 
equal  0,  and  (Art.  191)  the  section  will  be  a  parabola. 
When  A  =  0,  however,  tan2  d  =  tan2  <p  ;  or, 


That  is,  If  the  secant  plane  be  parallel  to  the  side  of  the 
cone,  the  section  will  be  a  PARABOLA. 

1.  If  we  suppose  d  —  tp  —  90°,  and  c  =  GO,  the  equa- 
tion to  the   section   can   readily  be  put   into   the   form 
y2  =  constant  ;  and  we  learn  that  when  the  vertex  of  the 
cone  recedes  to  infinity,  the  Parabola  breaks  up  into  two 
parallels. 

2.  If  d  =  (p,  and  c  —  0,  the  equation   to  the  section 
becomes  y2  =  0,  showing  that  the  limiting  case  of  the 
Parabola  is  a  right  line. 

III.  Let  A  be  negative.  The  function  H2  —  AB  will 
then  also  be  negative,  and  (Art.  174)  the  section  will  be 
an  hyperbola.  But  if  A  is  negative,  tan2  0  >  tan2  <p  ;  or, 


THE  CYLINDER  AND  THE  SPHERE.  567 

That  is,  If  the  inclination  of  the  secant  plane  be  greater 
than  that  of  the  side  of  the  cone,  the  section  will  be  an 

HYPERBOLA. 

1.  If  6  >  <p9  and  be  at  the  same  time  of  such  a  value 
that  (tan2  <p  —  tan2  6)  cos2  6  -f-  tan2  <p  =  Q9  the  equation  to 
the  section  will  satisfy  the  condition  A  +  B  =  0,  and 
the  section  (Art.  177)  will  be  a  rectangular  hyperbola. 

2.  If  0  >  y>9  and  c  —  0,  so  that  the  secant  plane  passes 
through  the  vertex,  the  section  becomes 

x2  (tan2  <p  —  tan2  6)  cos2  6  —  y2  tan2  <p  =  0, 

and  the  limiting  case  of  the  Hyperbola  appears  as  a 
pair  of  intersecting  right  lines. 

We  have  thus  shown  that  every  real  variety  of  the 
curve  of  the  Second  order  can  be  cut  from  a  right  cir- 
cular cone  by  a  plane.  The  imaginary  varieties,  of 
course,  can  not  be  obtained  by  any  geometric  process. 

736.  Equation    to   the   Circular    Cylinder.— The 

generatrix  of  this  surface  is  a  right  line  parallel  to  the 
axis :  hence,  if  we  take  the  latter  for  the  axis  of  z9  the 
generatrix  will  be  represented  by  x  =  a.  We  have,  then, 
f(z)  =  constant  =  a ;  and  the  required  equation  (Art.  732) 

x2  +  f  =  a2, 
in  which  a  is  the  radius  of  the  base. 

737.  [Equation  to  the  Sphere. — Taking   the  plane 
of  the  generating  circle  for  the  plane  ZX,  the  equation 
to  the  generatrix  is  x2  -f  z2  =  r2.     Hence, 


which  substituted  in  the  equation  of  Art.  732  gives 


as  the  equation  now  required. 
An.  Ge.  51. 


568  ANALYTIC  GEOMETRY. 

738.  Equations  to  the  Ellipsoids  of  Revolu- 
tion. —  Let  the  plane  of  the  generating  ellipse  be  the 
plane  XY.  The  equation  to  the  generatrix  will  be 

b2x2  -f-  d2y"  =  a2b2  :  whence 


Supposing  then  the  axis  major  to  be  the  axis  of  revo- 
lution, and  therefore  substituting  in  the  equation  for 
revolution  about  the  axis  of  x,  we  get 

—  ,  y1  +  ^  _  -i 

a2"1         V 

the  equation  to  the  Prolate  Spheroid. 

If  the  generatrix  lie  in  the  plane  ZX,  its  equation 
may  be  written  c2x2  -f-  a?z2  =  a?c2,  and  we  have 


Then,  taking  the  axis  minor  for  the  axis  of  revolution, 
we  substitute  in  the  equation  for  revolution  about  the 
axis  of  z,  and  obtain 

x2  +  y2    .   z2       .. 

~'  +   = 


the  equation  to  the  Oblate  Spheroid. 


fr.  Equations  to  the  Hyperboloids  of  Revolu- 
tion.— Changing  the  signs  of  b2  and  c2  in  the  expressions 
of  the  preceding  article  for  /  (x)2  and  /  (z)\  we  find 

/»    /        C  O          /       n  rt\  /i~~7 \  t*         /      o  0\ 


THE  HYPERBOLOID  AND  THE  PARABOLOID.  569 

Supposing  the  hyperbola  to  revolve  about  its  trans- 
verse axis,  we  substitute  in  the  equation  for  revolution 
about  the  axis  of  x,  and  obtain 

_  i 


_ 

a2  b* 

the   equation  to  the  Hyperboloid  of  Revolution  of  Two 
Nappes. 

Substituting  in  the  equation  for  revolution  about  the 
axis  of  2,  which  here  implies  that  the  hyperbola  revolves 
about  its  conjugate  axis,  we  get 

x2  +  y2       z2  _  , 
a2        "  c2  ~ 

the  equation  to   the  Hyperboloid  of  Revolution  of  One 

Nappe. 

74O.  The  Ellipse  of  the  Gorge.  —  This  name  is 
given  to  the  curve  cut  from  the  narrowest  part  of  the 
throat  of  ah  hyperboloid  of  one  nappe  by  a  plane  per- 
pendicular to  its  axis.  Its  equation,  found  by  putting 
z  =  0  in  the  equation  to  the  hyperboloid  (Art.  728,  II),  is 


In  the  Hyperboloid  of  Revolution,  this  curve  becomes 
the  circle  x2  +  y2  =  a2,  which  is  called  the  Circle  of  the 
Gorge. 

741.  Equation    to    the    Paraboloid    of   Revolu- 
tion. —  Beginning  with  the  generatrix  in  the  plane  XY, 

its  equation  is  y2  =  4px  :  whence 


f(x)=4px, 


570  ANALYTIC  GEOMETRY. 

which,  substituted  in  the  equation  for  revolution  about 
the  axis  of  x,  gives 

f  -f  z2  =  4px, 
the  required  equation  to  the  generated  surface. 

TANGENT  AND  NORMAL  PLANES  TO  THE  QUADRICS. 

742.  General  equation  to  the  Tangent  Plane.  —  By 

reverting  to  the  vectorial  equation  near  the  beginning  of 
Art.  718,  it  will  be  seen  that  the  radius  vector  will  meet 
the  Quadric  in  two  consecutive  points  at  the  origin,  if 
when  0  =  0  we  also  have 

6r  cos  a  -f  -Fcos  /?  -f-  D  cos  f  =  0. 

That  is  to  say,  multiplying  through  by  /?,  and  then  sub- 
stituting the  corresponding  x,  y>  2,  every  right  line  in  the 
plane 

Gx  +  Fy  -f  Dz  =  0 

is  a  tangent  to  the  surface  at  the  origin.  Hence,  the 
equation  just  written  is  the  equation  to  the  tangent  plane 
at  the  origin. 

Supposing  the  origin  not  to  be  on  the  surface,  by 
transforming  to  any  point  x'y'z'  of  the  surface,  the 
equation  to  the  tangent  plane  at  such  point  would  be, 
after  putting  for  the  new  G,  F,  D  their  values  as  given 
in  Art.  716, 


which,  by  re-transformation  to  the  original  axes,  gives 
(x  -  x')  Uxf  +  (y  —  y*)  UJ  -f  (z  -  z')  Uzf  =  0, 


TANGENT  AND  NORMAL  PLANES.  571 

as  the  general  equation  to  the  tangent  plane  at  any  point 
x'yV. 

Remark. — The  generic  interpretation  of  this  equation, 
as  of  its  analogue  in  the  Conies,  is  to  regard  it  as  the 
symbol  of  the  polar  plane  of  the  point  x'y'z',  which  in 
this  view  is  not  restricted  to  being  a  point  upon  the 
surface. 

743.  Tangent  Planes  to  the  different  Quad- 
rics. — The  equations  to  these,  in  their  simplest  forms, 
are  found  by  deriving  Uxf,  UJ,  Uxf  in  the  equations 
of  Arts.  728,  729,  and  substituting  the  results  in  the 
general  equation  last  obtained.  In  this  way,  we  get 

^  +  2/1  _|_  !!£  =  i, 

which  represents,  in  terms  of  the  semi-axes,  the  tangent 
plane  to  any  ellipsoid; 

?L?  _L  #V__^  —  1 
d2  '  '    b'2  "  "  c2  ~ 

which  represents  the  tangent  plane  to  any  Jiyperboloid  of 
one  nappe; 

x'x       y'y        z'z       ^ 

~o?   ~  '  ~V  '  ~  ~&~~      ' 

which  represents  the  tangent  plane  to  any  hyperboloid  of 
two  nappes; 


which  represents  the  tangent  plane  to  any  paraboloid. 

744.  Normal  of  a  Qiiadric. — The  right  line  per- 
pendicular to  any  tangent  plane  at  its  point  of  contact 


572  ANALYTIC  GEOMETRY. 

with  a  quadric,  is  called  the  normal  of  the  surface  at 
that  point.  Hence,  by  comparing  Arts.  709,  742,  we 

learn  that 

x  —  x' y  —  yr z  —  z' 

are  the  general  equations  to  the  normal  at  any  point  x'  y'  z'. 
By  deriving  Ux,  UJ,  UJ  in  the  equations  of  Art.  728, 
we   obtain  the  equations  to  the  normals  of  the  central 
quadrics,  namely, 

a2  (x  —  xf)  =  ±  b2(y  —  y')  =  ±  c2  (z  —  z'} 
x'  y'  z1 

in  which  we  must  use  the  upper  or  lower  signs  in  ac- 
cordance with  the  variety  of  the  surface.  If  we  derive 
Uxf,  Z7/,  Uzf  in  the  equation  of  Art.  729,  we  obtain 

x  —  x'      y'  —  y 


as  the  equations  to  the  normal  of  a  paraboloid.  These 
can  of  course  be  thrown  into  other  forms  when  con- 
venient. 

745.  Normal  Planes.  —  Any  plane  that  passes 
through  the  normal  of  a  quadric  at  any  point,  is  called 
a  normal  plane  to  the  quadric.  Comparing  Arts.  698 
and  744,  we  learn  that  the  general  equation  to  a  normal 
plane  will  be  of  the  form 

l(x-xf)        m(y-y'}  _    (I  +  m)  (z-Q 
UJ  UJ  Uz' 

where  the  arbitrary  k  of  Art.  698  is  for  the  sake  of 
symmetry  replaced  by  the  ratio  m  :  I. 

By  deriving  Ux'9  £/"/,  Uaf  from  any  specific  equation 
to  either  of  the  quadrics,  and  substituting  the  results  in 


EXAMPLES.  573 


the  preceding  formula,  we  can  obtain  the  equation  to  a 
normal  plane  for  any  given  quadric  in  any  given  system 
of  reference. 

It  is  noticeable  that  the  above  equation  involves  the 
indeterminate  ratio  m  :  I.  This  is  as  it  should  be  ;  for 
there  is  obviously  an  infinite  number  of  normal  planes 
corresponding  to  any  point  on  a  quadric.  When  the 
normal  plane,  however,  satisfies  such  conditions  as  de- 
termine it,  we  can  readily  find  the  corresponding  value 
of  in  :  Z. 

EXAMPLES  ON  THE  QUADRICS. 

1.  Determine,  by  means  of  their  discriminating  cubics  (see  Art. 
726),  whether  the  quadrics 

7#2  +  6?/2  +  5z2  —  4yz  —  4xy  =  6, 

Ix1  —  13y2  +  G22  +  24zy  +  I'lyz  —  Uzx  =  ±  84, 

2#2  +  3y2  +  4s2  +  6xy  +  4ijz  +  8zx  =  8, 

are  ellipsoids,  hyperboloids,  or  paraboloids. 

2.  In  any  central  quadric,  the  sum  of  the  squares  on  three  conju- 
gate semi-diameters  is  constant. 

3.  The  parallelepiped  whose  edges  are  three   conjugate  semi- 
diameters,  is  of  constant  volume. 

4.  Tangent  planes  at  the  extremities  of  a  diameter,  are  parallel. 

5.  The  length  of  the  central  perpendicular  upon  a  tangent  plane 
is  given  by  the  equation 


.    — 

p2        a*         b*          c4  ' 

6.  The  length  of  the  same  perpendicular,  in  terms  of  its  direction- 
cosines,  is 

p*  =  a2  cos2  a  -f  &2  cos2  /?  +  c2  cos2  7. 

7.  The  sum  of  the  squares  on  the  perpendiculars  to  any  three 
tangent  planes  is  constant. 


574  ANALYTIC  GEOMETRY. 

8.  The  locus  of  the  intersection  of  three  tangent  planes  which 
are  mutually  perpendicular,  is  the  sphere 


9.  Find  the  equation  to  a  diametral  plane  conjugate  to  a  fixed 
point  x'y'z';  and  prove  that  if  two  diameters  are  conjugate,  their 
direction-cosines  fulfill  the  condition 

cos  a  cos  of       cos  /?  cos  ft'       cosy  cos/  _ 
a  ~~r~  ~~~ 


10    The  locus  of  the  intersection  of  three  tangent  planes  at  the 
extremities  of  three  conjugate  diameters  is  the  central  quadric 


• 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


DEC  18  1947 

MAY  131948 


> 


mr 

j^ 

^\Bv 

Q^^ 


SAugSHt 

LD  21-100m-9,'47(A5702sl6)476 


FEB     ? 
AN  2  7  1955  LU 


U 


9Jan'58TSx 
EC'D  LO 

EC  18  19S7 


MAR  2  0  1S39 


-   LD 


JUN  1 


YU  ZZ370 


800547 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


